R&ESE  LIBRARY 

OF  THK 

UNIVERSITY  OF  CALIFORNIA. 

Deceived  ,  igo    . 

Accession  No.       92377...   Class  No. 


THE 


RAILWAY  TRANSITION  SPIRAL 


BY 

ARTHUR  N.  TALBOT,  C.  E. 

Member  American  Society  of  Civil  Engineers,  Prof  essor  of  Municipal 
and  Sanitary  Engineering.  University  of  Illinois 


THIRD    EDITION,    REVISED 


NEW  YORK  : 

ENGINEERING  NEWS  PUBLISHING  Co. 
1901 


Copyrig-ht,  1901, 

By 
ARTHUR  N.  TALBOT 


PREFACE. 


The  railway  transition  spiral  here  presented  is  a  flexible 
easement  curve  of  general  applicability  and  of  comparatively 
easy  analysis.  The  conceptions  and  methods  used  are  similar 
to  those  of  ordinary  circular  railroad  curves.  The  definition  is 
based  upon  degree-of-curve,  and  degree-of-curve,  central  angle, 
and  deflection  angle  may  be  calculated,  and  the  curve  may  be 
located  by  transit  and  chain  or  by  co-ordinates  from  tangent 
and  circular  curve.  The  field  work  is  quite  similar  to  that  for 
circular  curves.  The  spiral  is  easily  applied  to  a  variety  of  field 
problems  and  to  a  wide  range  of  location  and  old  track  condi- 
tions. In  the  principal  formulas,  angles,  co-ordinates,  offsets, 
etc.,  are  expressed  in  terms  of  the  length  or  distance  along  the 
spiral.  The  use  of  series  in  the  development  of  the  properties 
permits  an  estimate  of  the  error  involved  in  discarding  negligi- 
ble terms. 

The  principal  easement  curves  in  use  give  alignments 
which  approach  each  other  very  closely,  so  that,  for  equal 
easements,  it  may  be  expected  that  the  riding  qualities  will 
not  differ  sensibly.  In  general,  ease  of  calculation,  simplicity 
of  field  work,  and  general  applicability  and  flexibility  will 
determine  the  form  of  easement  curve  to  be  selected.  To 
establish  the  underlying  principles  of  an  easement  curve  of 
any  range  requires  considerable  mathematical  analysis.  The 
ordinary  treatment  of  circular  railway  curves  assumes  previous 
knowledge  of  the  geometrical  properties  of  the  circular  curve, 
but  the  properties  of  the  railway  spiral  must  be  deduced  from 
the  beginning.  Fortunately,  the  spiral  is  not  complex,  and  its 
properties  prove  to  be  simple  and  general.  A  general  treat- 
ment of  such  a  curve  has  many  advantages  over  approximate 
or  special  treatments.  Approximate  solutions  may  overlook 
important  variables,  and  short  methods  may  be  limited  to 
short  and  inefficient  easements.  The  range  of  conditions  of 
railway  curves  is  so  wide  that  it  is  best  to  develop  methods  of 
fairly  general  applicability,  and  these  may  then  be  simplified 
in  meeting  individual  conditions. 

92377 


The  treatment  herein  given  has  been  quite  widely  used  on 
the  railroads  of  the  United  States,  and  many  engineers  have 
commended  its  simplicity,  convenience,  and  flexibility.  The 
methods  and  principles  are  readily  taken  up  by  instrument 
men,  and  the  field  work  has  proved  little  more  difficult  than 
that  for  circular  curves.  The  use  of  a  regular  rate  of  transition 
per  100  feet  of  spiral  is  advantageous,  and  the  tables  are  in 
convenient  form. 

The  treatment  of  the  railway  transition  spiral  was  pub- 
lished in  Technograph  No.  5,  1890-91,  and  was  published  in 
field-book  size  in  1899.  Careful  attention  has  been  given  in 
this  revision  to  illustrative  examples  and  explanations.  The 
tables  have  been  extended  and  a  treatment  of  the  Uniform 
Chord  Length  Method  and  of  Street  Railway  Spirals  added. 
For  much  of  the  latter,  acknowledgment  is  made  to  Mr.  A.  L. 
Grandy.  The  writer  is  indebted  to  Messrs.  J.  K.  Barker,  Alfred 
L.  Kuehn,  and  many  others  for  valuable  assistance  in  the 
preparation  of  tables  and  text.  A.  N.  T. 

URBANA,  ILLINOIS, 

November  11,  1901. 


CONTENTS. 


Nomenclature , i 

Use.     Definition.     Notation.  Measurement  of  length. 

Theory 7 

Intersection  angle.  Co-ordinates  x  and  y.  Spiral 
deflection  angle.  Table  of  corrections.  Deflection 
angle  at  point  on  spiral.  Ordinates.  Offset.  Ab- 
scissa of  P.C.  Tangent-distance.  External-distance. 
Long  chord.  Spiral  tangent-distances.  Middle  ordinate. 

Summary  of  Principles 20 

Principal  formulas.  Angles  and  deflection  angles. 
Angles  from  tangent.  Angles  from  chord.  Diverg- 
ence from  osculating  circle.  Co-ordinates.  Offset. 
Other  distances. 

Description  and  Use  of  the  Tables 28 

Tables  for  transition  spiral.  Accuracy.  Interpola- 
tion. General  use  of  Table  IV.  Corrections  for  cal- 
culations. Other  tables. 

Choice  of  a  and  Length  of  Spiral 32 

Effect  of  speed.  Attainment  of  superelevation. 
Amount  of  superelevation.  Minimum  spiral.  Selec- 
tion of  spiral. 

Location  of  P.S.,  P.C.C.,  and  P.C 37 

Laying  out  the  Spiral  by  Co-ordinates 39 

From  tangent,  tangent  and  curve,  and  spiral  tangents. 

Location  by  Transit  and  Deflection  Angles 42 

Transit  at  P.S.  Transit  on  spiral.  Intermediate  de- 
flection angles.  Transit  at  P.C.C.  Transit  notes. 

Application  to  Existing  Curves 49 

To  replace  the  entire  curve—  Two  methods.  To  re- 
place part  of  the  curve.  To  re-align  and  compound. 
Methods  of  trackmen. 


Compound  Curves  — 59 

General  method.    To  insert  in  old  track. 

Miscellaneous  Problems 66 

To  change  tangent  between  curves  of  opposite  direc- 
tion. To  change  tangent  between  curves  of  same 
direction. 

Uniform  Chord  Length  Method 71 

Formulas.  Tables.  Fractional  chord  lengths.  Use 
of  method. 

Street  Railway  Spirals 80 

Theory.  Tables.  Laying  out.  Arc  excess.  Curving 
rails.  Double  track. 

Conclusion 88 

Explanation  of  Tables 92 

Tables  I-XI.    Transition  Spirals 94-105 

Table  XII.     Factors  for  Ordinates 105 

Table  XIII.     Unit  Spiral  Deflection  Angles 106 

Table  XIV.    Coefficients  for  Deflection  Angles 107 

Tables  XV-XIX.     Street  Railway  Spirals 108-109 

Table  XX.    Offsets  for  Spirals  no 


THE  RAILWAY  TRANSITION  SPIRAL 


NOMENCLATURE 

1 .  A  transition  curve,  or  easement  curve,  as  it 
is  sometimes  called,  is  a  curve  of  varying-  radius 
used  to  connect  circular  curves  with  tangents  for 
the  purpose  of  avoiding  the  shock  and  disagreeable 
lurch  of  trains  due  to  the  instant  change  of  direc- 
tion and  also  to  the  sudden  change  from  level  to 
inclined  track.  By  this  means  the  superelevation 
of  the  outer  rail  may  be  made  to  correspond  to  the 
curvature  at  all  points  around  the  transition  curve. 
The  primary  object  of  the  transition  curve,  then,  is 
to  effect  smooth  riding  when  the  train  is  entering 
or  leaving  a  curve. 

The  generally  accepted  requirement  for  a  proper 
transition  curve  is  that  the  degree-of-curve  shall 
increase  gradually  and  uniformly  from  the  point  of 
tangent  until  the  degree  of  the  main  curve  is 
reached,  and  that  the  super-elevation  shall  increase 
uniformly  from  zero  at  the  tangent  to  the  full 
amount  at  the  connection  with  the  main  curve  and 
yet  have  at  every  point  the  appropriate  super- 
elevation for  the  curvature.  In  addition  to  this,  an 
acceptable  transition  curve  must  be  so  simple  that 
the  field  work  may  be  easily  and  rapidly  done,  and 
should  be  so  flexible  that  it  may  be  adjusted  to  meet 
the  varied  requirements  of  problems  in  location  and 
construction. 


2  NOMENCLATURE 

No  attempt  will  here  be  made  to  show  the  neces- 
sity or  the  utility  of  transition  curves.  The  prin- 
ciples and  some  of  the  applications  of  one  of  the 
best  of  these  curves,  the  railway  transition  spiral, 
will  be  considered. 

2.  Definition.  The  Transition  Spiral  is  a  curve 
-whose  degree-of-curve  increases  directly  as  the  dis- 
tance along  the  curve  from  the  point  of  spiral. 

Thus,  if  the  spiral  is  to  change  at  the  rate  of  10° 
per  100  feet,  at  10  feet  from  the  beginning-  of  the 
spiral  the  curvature  will  be  the  same  as  that  of  a  1° 
curve;  at  25  feet,  as  of  a  2°30'  curve;  at  60  feet,  as 
of  a  6°  curve.  Likewise,  at  60  feet,  the  spiral  may 
be  compounded  with  a  6°  curve;  at  80  feet,  with  an 
8°  curve,  etc. 

This  curve  fulfills  the  requirements  for  a  transi- 
tion curve.  Its  curvature  increases  as  the  distance 
measured  around  the  curve.  The  formulas  for  its 
use  are  comparatively  simple  and  easy.  The  field 
work  and  the  computations  necessary  in  laying  it 
out  and  in  connecting  it  with  circular  curves  are 
neither  long  nor  complicated,  and  are  similar  to 
those  for  simple  circular  curves.  The  curve  is 
extremely  flexible,  and  may  easily  be  adapted  to  the 
requirements  of  varied  problems.  The  rate  of 
change  of  degree-of-curve  may  be  made  any  desira- 
ble amount  according  to  the  maximum  curve  used, 
the  maximum  speed  of  trains  or  the  requirements 
of  the  ground. 

As  the  derivation  of  the  formulas  is  somewhat 
long,  their  demonstration  will  be  given  first.  The 
explanation  and  application  of  these  formulas  to 
the  field  work  and  to  the  computations  will  be  given 


NOMENCLATURE  3 

separately,  a  knowledge  of  the  demonstration  not 
being"  essential  to  the  application. 

3,  In  Fig-.  1,  DLH  is  the  circular  curve  and  AP 
the  prolongation  of  the  initial  tangent  which  are  to 
be  connected  by  the  transition  spiral.     D  is  the  point 
where  the  completed  circular  curve  gives  a  tangent 
DN  parallel  to  the  tangent  AP,  and  will  be  called 
the  P.C.  of  the  circular  curve.     AEL  is  the  transi- 
tion spiral  connecting  the  initial  tangent  AP  with 
the  main  or  circular  curve  LH.     A  is  the  beginning 
of  the  spiral  and  will  be  known  as  P.S.,  point  of 
spiral.     L  is  the   beginning  of  the  circular  curve 
LH,  and  will  be  called   P.C.C.,    point  of  circular 
curve.     AP  will  be  used  as  the  axis  of  X,  and  A  as 
the  origin  of  co-ordinates.    BD  is  the  offset  between 
the  tangent  AB  of  a  circular  curve  and  spiral,  and 
the  parallel  tangent  DN  of  an  unspiraled  curve. 

The  degree-of-curve  of  the  spiral  at  any  point  is 
the  same  as  the  degree  of  a  simple  curve  having  the 
same  radius  of  curvature  as  the  spiral  has  at  that 
point.  The  radius  of  the  spiral  changes  from 
infinity  at  the  P.S.  to  that  of  the  main  curve  at  the 
P.C.C.  The  spiral  and  a  simple  curve  of  the  same 
degree  will  be  tangent  to  each  other  at  any  given 
.point;  i.  £.,  they  will  have  a  common  tangent. 

4.  The  following  notation  will  be  used: 
P.S.  ==  Point  of  spiral.     (A,  Fig.  1.) 

p.C.C.  —  point  where  spiral  compounds  with  cir- 
cular curve.  (L,  Fig.  1.) 

P.C.  =  beginning  of  offsetted  circular  curve.  (D, 
Fig.  1.) 

R  =  radius  of  curvature  of  the  spiral  at  any  point. 

D  =  degree-of-curve  of  the  spiral  at  any  point; 


4  NOMENCLATURE 

sometimes  called  D\  at  the  end  of  spiral.  Gener- 
ally D\  is  made  the  same  as  Z>o,  the  degree  of  the 
main  curve. 

a  =  rate  of  change  of  the  degree-of-curve  of  the 
spiral  per  100  ft.  of  length.  It  is  equal  to  the 
degree-of-curve  of  the  spiral  at  100  ft,  from  the  P.S. 

5  =  length  in  feet  from  the  P.S.  along-  the  curve 
to  any  point  on  the  spiral. 

L  —  number  of  100-ft.  stations  from  the  P.S.  along 
the  curve  to  any  point  on  the  spiral;  in  other  words 
the  distance  to  any  point  measured  in  units  (or  sta- 
tions) of  100ft.  For  the  whole  spiral  (to  P.C.C.) 
it  is  sometimes  called  L\. 

1=  total  central  ang-le  of  the  whole  curve  (inter- 
section angle),  or  twice  BCH  of  Fig-.  1,  H  being-  the 
middle  of  the  circular  arc. 

J  —  ang-le  showing-  the  chang-e  of  direction  of  the 
spiral  at  any  point,  and  is  the  ang-le  between  the 
initial  tangent  and  the  tang-ent  to  the  spiral  at  the 
g-iven  point.  For  the  whole  spiral  it  is  equal  to 
PTL  and  may  be  called  A.  The  latter  is  also  equal 
to  DCL. 

9  =  spiral  deflection  ang-le  at  the  P.S. ,  from  the 
initial  tangent  to  any  point  on  the  spiral.  For  the 
point  L  (Fig.  1)  it  is  BAL. 

$  =  deflection  angle  at  any  point  on  the  spiral, 
between  the  tangent  at  that  point  and  a  chord  to 
any  other  point.  At  L,  for  the  point  A,  0  is  TLA. 
x  =  abscissa  of  any  point  on  the  spiral,  referred 
to  the  P.S.  as  the  origin  and  the  initial  tangent  as 
the  axis  of  X.  For  the  point  L,  oc  =  AM. 

y  =  ordinate  of  the  same  point,  measured  at  right 
angles  to  the  above  axis.  For  the  point  L,  y  =  ML. 


NOMENCLATURE 


5 


/—  abscissa  of  the  P.C.  of  the  main  curve  produced 
backward:  z.  <?.,  of  a  simple  curve  without  the  spiral. 
For  P.C.  at  D,  /=AB. 

o  =  offset  between  the  initial  tangent  and  the  par- 
allel tangent  from  the  main  curve  produced  back- 
ward, or  it  is  the  ordinate  of  the  P.C.  of  the  pro- 
duced main  curve.  If  D  is  the  P.C.,  BD  is  o.  It  is 
also  the  radial  distance  between  the  concentric  cir- 
cles LH  and  BK. 

7"=  tangent-distance  for  spiral  and  main  curve 
=  distance  from  A  to  the  intersection  of  tangents. 

E  =  external-distance  for  spiral  and  main  curve. 

C=long  chord  AL  of  the  transition  spiral. 

u  =  distance  along-  initial  tangent  from  P.S.  to 
intersection  with  spiral  tangent  =  AT  for  point  L. 

v  =  length  of  spiral  tangent  to  intersection  with 
initial  tangent  =  TL  for  point  L. 

5.  The  length  of  the  spiral  is  to  be  measured 
along  chords  around  the  curve  in  the  same  way  that 
simple  curves  are  usually  measured,  using  any 
length  of  chord  up  to  a  limit  which  depends  upon 
the  degree-of-curve  of  the  spiral.  The  best  railroad 
practice,  in  the  writer's  opinion,  considers  circular 
curves  up  to  a  7°  curve  as  measured  with  100-ft. 
chords,  from  7°  to  14°  as  measured  with  50-ft. 
chords,  and  from  14°  upwards  as  measured  with 
25-ft.  chords;  that  is  to  say,  a  7°  curve  is  one  in 
which  two  50-ft.  chords  together  subtend  7°  of  cen- 
tral angle,  a  14°  curve  one  in  which  four  25-ft. 
chords  together  subtend  14°  of  central  angle.  The 
advantages  of  this  method  are  two-fold, — the  length 
of  the  curve  as  measured  along  the  chords  more 
nearly  approximates  the  actual  length  of  the  curve, 


6  THEORY 

and  the  radius  of  the  curve  is  almost  exactly 
inversely  proportional  to  the  degree-of-curve.  The 
latter  consideration  is  an  important  one,  simplifying 
many  formulas.  With  this  definition  of  degree-of- 
curve,  the  formula  R  =  ~~^~  will  give  no  error  great- 
er than  1  in  2  500.  For  a  10°  curve  the  error  in  the 
radius  is  .15  feet,  and  for  a  16°  curve  .06  feet.  This 
approximate  value  of  R  will  give  a  resulting  error 
in  the  length  of  the  spiral,  for  the  ordinary  limits 
of  spirals,  of  less  than  f^  of  the  length,  and  will 

A  B  T  M 


FIG.   1. 

not  reach  0.1-ft.  The  resulting*  error  in  alignment 
is  6-^Vo  y*  anc^  wi^  not  reach  0.01-ft.  The  difference 
between  the  length  of  the  curve  and  that  of  these 
chords  is  less  than  1  in  7  000.  For  spirals  measured 
with  lengths  of  chords  as  here  specified,  or  shorter, 
the  error  either  in  alignment  or  distance  will  be 
well  within  the  limits  of  accuracy  of  the  field  work, 
and  hence  the  relation  R=^r  will  be  considered 
true. 


INTERSECTION  ANGLE  7 

THEORY 

6.  Intersection  Angle  ^.  —  From  the  definition  of 
the  transition  spiral,  we  have,  remembering  that  the 
value  of  a  as  defined  above  requires  the  length  of 
curve  to  be  measured  in  100-ft.  units  (stations} 
instead  of  feet,  H/tfTJt£&^J 

D  =  aL=as   .........................  (1) 

100 

For  the  P.C.C.  this  becomes  D\  =  aL\. 
From  the  calculus  the  radius  of  curvature 
R=  ds 


5  T  3  0  ,      . 

Substituting-  the  expression  R=  ~^~  and  solving, 

7  A        a  s  ds 
a  A  = 


573000 

T    .          , .  a  s2  a  L? 

Inte^ratlD§r'J=  1146000  =  TT^6 
Changing  A  from  circular  measure  to  degrees, 


(2) 


which  is  the  intrinsic  equation  of  the  Transition 
Spiral. 

For  the  P.C.C.  this  becomes  Ji  =  ^  aL-f. 


Since  from  (1)  a  =  -,  we  also  have 


(3) 


From  these  equations  it  will  be  seen  that 

(a)  the  change  of  direction  of  the  spiral  varies  as 
the  square  of  the  length  instead  of  as  the  first  power 


8  THEORY 

of  the  length  as  in  the  simple  circular  curve,  and 

(b)  that  the  transition  spiral  for  any  angle  A  will 
be  twice  as  long  as  a  simple  circular  curve. 

7.  Co-ordinates^  and  y.  To  find  the  co-ordinates, 
x  and  y,  of  any  point  on  the  spiral,  we  have  by  the 
calculus  dy  =  ds  sin  A  and  dx=  ds  cos  A.  Expand- 
ing the  sine  and  cosine  into  an  infinite  series,  sub- 
stituting for  ds  its  value  in  terms  of  dA,  and  inte- 
grating, we  have 

1070.51^     |       iji,      ,     jV_etc.  1(4) 
(«)*    \/          -*        f™  fW 


As  ^  here  is  measured  in  circular  measure  and 
is  only  >£  when  the  angle  is  28°. 65,  these  series  are 
rapidly  converging,  especially  for  smaller  angles. 

Changing  the  angle  A  from  circular  measure  to 
degrees,  substituting  for  J,  and  dropping  the  small 
terms, 

y  =-.291  aU  —  .00000158  a3  D (6) 

For  values  of  A  less  than  15°  the  last  term  may  be 
dropped,  and  up  to  25°  the  term  will  be  small. 
D  L?  may  also  be  written  in  place  of  a  L?.  For  all 
except  extreme  lengths,  the  last  term  may  be 
dropped.  Using  y  =  .291  a  U,  it  is  seen  that  y 
varies  as  the  cube  of  the  distance  of  the  point  from 
the  P.S. 

Likewise  changing  A  from  circular  measure  to 
degrees,  etc., 

x  =  100  L— .  000762  a2  U +.  0000000027  0*  U . .  ( 7) 
Or  oc=  100  L-. 000762  D*  U (8) 


UNIVERSITY 

V  OF 

SPIRAL 


The  second  term  in  second  member  of  equation 
(7)  or  (8)  may  be  used  as  a  correction  to  be  sub- 
tracted from  the  length  of  the  curve  in  feet.  The 
last  term  in  equation  (7)  can  be  omitted,  except 
for  extreme  lengths. 

8  Spiral  Deflection  Angle  #.—  It  is  desired  to  find 
the  deflection  angle  9  for  any  point  on  the  spiral,  as 
BAL  for  the  point  L  (Fig-.  1.)  To  show  that  this 
is  nearly  ^  ^,  divide  equation  (4)  by  equation  (5). 

tan  9  =  Yi  A  +  Tfo  J*  +  „#„  *>,  etc.  But  from 
the  tangent  series  for  y$  A, 

tan  #  J  =  Yz  A  +  ^  A*  +  TlfrT  J5,  etc.  Subtracting- 
one  from  the  other,  we  get  a  series  which  is  rapidly 
decreasing-  when  A  is  less  than  40°.  Investigating 
this  difference,  remembering-  that  A  is  in  circular 
measure,  it  is  found  that  the  error  of  calling"  the 
two  equations  equal  is  less  than  1'  for  A  =  25°  and 
decreases  rapidly  below  this.  As  A  will  rarely 
reach  25°,  and  is  only  a  small  fraction  of  a  minute 
for  any  angle  ordinarily  used,  and  as  the  resultant 
error  of  direction  will  be  corrected  at  the  P.C  C. 
when  A  —  9  is  turned  off,  we  may  ordinarily  dis- 
reg-ard  this  and  write 

#  =  /3^=-^Z2  =  i— W 

a 

where  0  is  in  degrees. 

From  equation  (9)  it  is  seen  that  the  spiral  de- 
flection angles  to  two  points  on  the  spiral  will  be  to 
each  other  as  the  square  of  the  distances  to  the 
points. 

9.  The  error  in  equation  (9)  is  dependent  upon 
the  value  of  A  or  9  and  hence  may  be  expressed 


THEORY 


independently  of  the  length  of  spiral  and  rate  of 
transition.  For  A  between  20°  and  40°,  the  number 
of  minutes  correction  to  be  subtracted  from  \  A  or 
\  a  U  to  give  9  is  .000053  A*  where  A  is  in  degrees. 
The  following1  table  gives  the  deductions  for  various 
angles,  and  for  other  values  interpolations  may  be 
made: 


Correction  in  minutes  to  be  subtracted  from 
J-  a  U  to  give  more  precise  values  of  6. 


A  or 


A 

12° 
15° 
18C 


Cor. 
0.1 
0.2 
0.3 


A 
21C 

24C 
27° 


Cor. 
0.5 
0.7 
1.0 


A 

30° 
33° 
36° 


Cor. 
1.4 
1.9 

2.4 


°,  the  real 
J'        F  B 

value  of 
G 

9  will  be 

T       M 

:5^4 

""  -K  .. 
Dl 

^^sJ8 

\  A     ! 

V' 

\ 
\ 

*         !!l 

1  """' 

^S)ft> 

-    T-  --,U 
\        i 

%.   \    \ 

>» 

fC'      x>% 

*c  r> 

it- 

/               x        \ 

c^-^ 

mtip 

>\\\r- 

\»  A'NlH 

WN 

FIG.  2. 

66  —  0.'3  =  5°59.'7.  For  a  value  of  ^  near  6°  the 
same  correction  may  be  made.  It  will  be  seen  that 
for  the  spiral  deflection  angles  ordinarily  used  the 


DEFLECTION  ANGLE  n 

correction  may  be  neg-lected  without  material  error. 

For  the  terminal-point  of  the  spiral,  the  P.C.C., 
the  value  of  #1  may  be  obtained  from  equation  (9). 
In  the  extreme  cases,  where  a  further  term  is  needed, 
the  correction  may  easily  be  made  from  the  above 
table. 

10  Spiral  Tangent  — To  find  the  tang-ent  at  any 
point  of  the  spiral,  L,  lay  off  a  deflection  ang-le  from 
LA  equal  to  1  —  0.  When  J  is  not  over  20°,  2/z  1  or, 
2  9  may  be  used.  This  since  FLT  =  PTL  =-  J,  and 
FLA  ==  PAL  =  9.  This  is  true  for  any  point. 

For  the  terminal  point  of  the  spiral,  P.C.C.,  this 
becomes  Ji  —  Q\  which  is  g-enerally  expressed  with 
sufficient  precision  by  2/$  Ji. 

1 1 .  Deflection  Angle  at  Point  on  Sp'rral. — The  de- 
flection ang-le  from  the  tang-ent  at  any  point  on  the 
spiral  to  locate  a  second  point  may  be  found  as  fol- 
lows: In  Fig-.  2,  let  L'  be  the  distance  from  the 
P.S.  to  R,  and  L  the  distance  from  the  P.S.  to  any 
other  point  on  the  spiral,  as  K.  Let  FRN  be  the 
tang-ent  at  R,  and  RFM—  1'  its  ang-le  with  the 
initial  tang-ent,  and  9'  the  corresponding-  spiral  de- 
flection ang-le,  RAF.  Let  KTM=  J  be  ang-le  of 
tang-ent  at  K  with  initial  tang-ent,  equal  to  total 
chang-e  of  direction  of  the  spiral  up  to  that  point. 
9f  and  9  are  the  deflection  ang-les  at  the  P.S.  for  R 
and  K  respectively.  KRN  =  $  =  required  deflection 
ang-le.  KRU  =  0-f  A'. 

To  show  that  the  ang-le  $  -\-  A'  is  almost  exactly 

3  3 

A  *  —  J '  * 

the  same  as  the  ang-le  yz  — \ —     — \  or  \  (l-\-A*A'*  +^')> 
the  following"  somewhat  long-  and  tedious  operation 


12  THEORY 


may  be  gone  through.     It  is  thought  not  necessary 
to  give  it  in  detail  here. 


RU      #-#1 

Substitute  for  the  co-ordinates  in  the  above  equation 
their  values  from  equations  (4)  and  (5),  and  also 

develop  tan   ^  — —    — j-  into  a  series,  and  subtract 
•A  *  —  A** 

the  latter  from  the  former.  An  expression  for  the 
difference  will  be  found,  which  amounts  to  but  a 
small  fraction  of  a  minute  for  any  value  of  A  up  to 
35°.  Hence  we  may  write 

(p  4-  A'  =  V*  (A  4-  A*  A'*  4-  A'^\ 

i  /  -3  \       \  j 

By  substituting  for  A  and  A'  their  values  in  terms 
of  L  and  L'  and  reducing,  the  following  value  for 
$  is  found: 

Also,J'  — 0'db0=0  +  iZ>'Z (11) 

And   zT:4=0=:6>/+<9  +  \D' L (12) 

Even  for  very  large  angles  these  equations  are 
quite  accurate  if  the  exact  value  of  0  is  used.  In 
equation  (10)  the  last  term  %  a  (L — Z')2  should 
receive  the  same  correction  as  an  equal  value  of  6. 
Of  course,  for  any  angle  ordinarily  used  no  correc- 
tion need  be  made.  See  correction  for  0,  page  10. 

12.  In  equation  (10)  it  will  be  noticed  that  the 


DEFLECTION  ANGLE  13 

first  term  (\a  L'  (L— Z/))  is  equal  to  the  deflection 
angle  for  a  simple  circular  curve  of  the  same  degree 
as  the  spiral  at  the  point  R  (z.  £.,  a  L'')  and  of  a 
length  equal  to  the  distance  between  the  two  points; 
while  the  second  term  (-J-  a  (L—L'  )2)  is  equal  to  the 
spiral  deflection  angle  at  the  P.S.  from  the  initial 
tangent  for  an  equal  length  of  spiral  (Z— Z' ). 

If  the  point  to  be  located  had  been  chosen  on  the 
side  of  R  nearer  to  the  P.S.,  the  two  terms  of  equa- 
tion (10)  would  have  opposite  algebraic  signs,  and 
the  difference  of  the  two  quantities  would  be  used. 
To  show  that  the  arithmetical  difference  of  the 
two  terms  is  to  be  used  for  a  point  nearer  the  P.S. 
when  the  distance  (L — Z')  is  used  without  regard 
for  the  algebraic  sign,  equation  (10)  has  been  writ- 
ten with  the  plus  and  minus  sign. 

13,  The  spiral  then  deflects  from  a  circle  of  the 
same  degree-of-curve  at  the  same  rate  that  the  spiral 
deflects  from  the  initial  tangent  at  the  beginning. 
D'RH,  in  Fig.  2,  represents  the  circular  curve  tan- 
gent to  spiral  at  R,  the  two  having  the  same  radius 
at  that  point  and  both  being  tangent  to  FRN.  The 
deflection  angles  between  points  on  the  spiral  and  on 
the  circle  RH,  and  also  between  the  spiral  and  RD'  are 
the  same  as  the  spiral  deflection  angle  for  an  equal 
length  of  spiral  from   A.     In  the  same  way  at  K, 
RKT^SKT  —  SKR,  the  latter  angle  being  equal 
to  the  deflection  from  initial  tangent  at  A  for  a 
length  of  spiral  equal  to  KR. 

14.  Equation  (11)  shows  that  the  angle  at  any 
point  between  the  chord  joining  this  point  with  the 
P.S.  and  a  chord  to  any  other  point   (the   angle, 
Fig.  2,  between  AR  produced  and  RK  if  the  point 


14  THEORY 

K  is  to  be  located  from  R)  is  equal  to  the  spiral  de- 
flection angle  at  the  P.S.,  0,  for  the  point  to  be 
located  (KA.M)  plus  one  third  of  the  deflection 
angle  for  a  circular  curve  of  the  same  degree  as  that 
of  the  spiral  at  the  vertex  of  the  angle,  R,  and  of 
the  length  of  the  spiral  from  P.S.  to  the  point  K, 
This  is  true  whether  the  point  to  be  located  is 
nearer  the  P.S.,  or  farther,  than  the  point  used  as 
the  vertex  of  the  angle. 

It  may  also  be  readily  shown  from  (2)  that  the 
difference  in  direction  of  the  two  tangents,  A — J', 
is  the  central  angle  for  this  simple  curve  plus  the 
spiral  angle,  both  for  a  length  equal  to  the  distance 
between  the  two  points. 

15  Equation  (12)  gives  the  value  of  the  deflec- 
tion angle  from  a  line  parallel  to  the  initial  tangent, 
the  spiral  deflection  angle  0'  for  the  point  R  being 
added  to  the  values  in  equation  (11), 

16  Ordinaies   from    osculating    circle. — It    may 
also   be   shown  that  the  offset  distance  between  a 
point  on  the  spiral  and  one  on  the  osculating  circle 
is  the  same  as  the  ordinate  y  from  the   initial  tan- 
gent at  a  point  the  same  distance  from  the  P.S.  as 
the  former  point  is  from  the  point  of  osculation. 
These  ordinates  may  be  measured   in   a  direction 
normal  to  the  circular  curve. 

17.  Offset  o.~ From  Fig.  1,  BD  =  BF— DF  = 
BF  — CD  vers  DCL.  But0=BD,  BF  =  y  for  the 
end  of  spiral,  DCL=^  for  the  whole  spiral,  and 
CD  =  jR.  Hence,  o=y  —  R  vers  J.  Substituting 
f or  y,  R,  and  ^  their  values  in  terms  of  the  length 
of  the  whole  spiral,  applying  the  versed  sine  series, 
and  reducing  we  have  for  o  in  feet 


OFFSET  GIVEN,  ETC.  15 

o  =  .0727  a  Lis  =  .0727  Di  L?  .............  (13) 

where  D\  and  L\  refer  to  the  whole  length  of  the 
spiral.  The  other  terms  of  the  series  are  so  small 
that  they  may  be  dropped  when  A  is  less  than  30°. 
The  'next  term  is—  .0000002  a*  D.  It  will  be  seen 
that  o  is  approximately  one  fourth  of  the  ordinate 
of  the  P.C.C.,  which,  of  course,  should  be  true  if  E, 
the  middle  point  of  the  spiral,  is  opposite  D,  the  P.C. 

18,   Offset  given.  —  From  (11)  and  (3)  we  have 


(14) 
J  =  1.857  V"^  ....................... 


3|,  l^  and  fa  maJ  be  used  for  these  co-efficients  with 
advantage. 

19.  Abscissa  of  P.O.,  t.—  From  Fig.  1,  /==  AB 
==  AM  —  BM  =  x  —  FL  —  x  —  R  sin  J.  Expanding 
and  reducing, 

/  =  50  Zi—  .000127  a*  Lf\ 
or  >  ..............  U'.) 

/  =  50  Li  -  .000127  D?  L?  j 

It  should  be  noted  that  the  full  length  of  the 
spiral  is  used  in  the  formula.  The  last  term  may 
be  used  as  a  correction  tc  be  subtracted  from  the. 
half  length  of  the  spiral.  It  is  easily  tabulated  for 
the  principal  spirals,  and  corrections  for  other 
spirals  may  be  found  by  multiplying  the  value  with 
a  =  I  for  the  given  length  of  spiral  by  the  square 
of  the  a  used. 


16  THEORY 

20.  A  comparison  of  /  with  the  abscissa  found  by 
substituting-  >^  L\  in  equation  (8)   shows  that  BD 
cuts  the  spiral  at  a  point  only  .0001  a2  U  feet  from 
the  middle  point  of  the  spiral.     This  is  f  of  the  cor- 
rection used  in  equation  (17)  for  finding-  /  from  ^  Za. 
For  our  purpose  we  may  say  that  BD  bisects  the 
spiral.     It  also  follows  that  the  spiral  bisects  the 
line  BD,  since  BE  =  %y*     This  is  subject  to  slight 
error  for  larg-e  ang-les. 

The  length  of  the  spiral  from  the  P.S.  to  BD, 
therefore,  exceeds  /  by  one  fifth  of  the  t  correction, 
and  the  remainder  of  the  spiral  exceeds  the  length 
of  the  circular  curve  from  the  P.C.  to  the  P.C.C.  by 
four  fifths  of  the  t  correction.  The  entire  length  of 
the  spiral  exceeds  the  distance  measured  on  t  (AB, 
Fig*.  1)  plus  the  distance  measured  around  the  cir- 
cular curve  (DL,  Fig1.  1)  by  the  t  correction  g-iven 
in  equation  (17), 

21.  Tangent-distance  T.—  To  find  T,  consider  in 
Fig.  1  that  AB  intersects  CH,  H  being-  the  middle 
of  the  circular  curve,  at  some  point  P  outside  the 
diagram.      Then  T  =  AP  =  AB  +  BP.      BP  =  BC 
tan  BCH. 

Hence  T=t+(R+  0)  tan  }£/.... (18) 

t  and  o  tan  J^  /may  be  computed  separately  and 
added  to  the  T  found  from  an  ordinary  table  of 
tangent-distances. 

22.  Equation  (18)  gives  T  for  the  same  transition 
spirals  at  each  end  of  the  main  curve.     It  may  be 
desirable   to   make   one   spiral   different   from   the 


TANGENT  DISTANCE  17 

other.  To  find  an  expression  for  the  tangent- 
distances  for  this  case  proceed  as  follows:  In  Fig. 
3,  let  RS  =  HD  =  o2,  BD  =  01,  AB  =  /i,  RT  =  /2,  AE 
=  71,  TE=  7},  R=  radius  of  main  curve  DLKS, 
R-\-  02  =  radius  of  HR,  and  J=  angle  PER. 


FIG.  3. 
Then  71  =  h  +  HC  —  PE,  and 

7i  =  A  +(/?  +  02)  tan  Y*  I—  (01  —  o^coil.  .(19) 

Similarily,  7^2  —  /2  +  (/?  +  02)  tan  >^  7+  (01  —  02) 
cosec  /. 

When  /  is  more  than  90°,  the  last  term  of  (19) 
becomes  essentially  positive. 


l8  THEORY 


23.  External-distance  E.  —In  Fig.  1,    E  = 
HK.     Hence 


E=  (JR  +  0)  exsec  tff+o  .........  _____  (20) 


24.  Long  chord    C.—  In  Fig-.  1,   C=  AL.  ML  = 

ALsinMAL,  orC=-r^---     Putting  this  in  terms 
sin  0 

of  the  length  of  the  curve, 

C=  100  Z—  (.000338  c?D  or  .000338  Z>2£3)  .  (21) 


in  which  C  is  in  feet  and  L  in  stations.  It  will  be 
seen  that  the  last  or  correction  term  is  four  ninths 
of  the  correction  for  x  as  given  in  equation  (7). 
When  the  correction  term  for  x  is  tabulated  or 
otherwise  known,  the  length  of  the  long  chord  may 
conveniently  be  calculated  by  subtracting  four 
ninths  of  this  x  correction  term  from  the  whole 
length  of  the  spiral. 

25.  Spiral  tangent-distances.  —  In  Fig.  1,  u  =  A.T 
=  AM  —  MT.     As  MT  =y  cot  J  or  v  cos  J, 

u  =  x  —  y  cot  ^  ........................  ] 

I,  (22) 
u*=*x  —  v  cos  A  .......................    j 

Also  v  —  TL,  '=  •  .     4  .     Expanding-  sin  J  into  series, 
sin  a 

substituting  the  value  of  y  from  equation  (6),  and 
reducing, 


v  _  --=        .Z  +  .000244  ^  Z5  .........  (23) 

sin  J         3 


MIDDLE    ORDINATE  19 

The  last  term  is  almost  exactly  one  third  the  cor- 
responding- term  in  equation  (7),  and  hence  v  may 
be  found  by  adding  one  third  of  the  correction  term 
used  for  determining  x  to  one  third  of  the  length  of 
the  spiral  in  feet. 

26.  Middle  ordinate. — The  middle  ordinate  for 
any  arc  of  the  spiral  is  equal  to  the  middle  ordinate 
for  an  equal  length  of  circular  curve  of  the  same 
degree-of-curve  as  the  spiral  at  the  middle  point  of 
the  arc  considered.  This  degree-of-curve  is  the 
mean  of  the  D"§  at  the  end  of  the  given  arc. 
This  is  an  approximate  formula  which  is  true 
whether  one  end  of  the  chord  is  at  the  P.S.  or  not. 

The  ordinate  from  any  other  point  along  a 
chord  may  be  found  as  follows:  Since  the  spiral 
diverges  from  the  osculating  circle  at  the  middle 
point  of  the  arc  at  the  same  rate  as  from  the  initial 
tangent,  the  amount  of  this  divergence  may  be  cal- 
culated by  the  method  given  oh  page  14  and  added 
to  or  subtracted  from  the  ordinate  for  the  osculat- 
ing circular  curve.  For  a  point  nearer  the  P.  S. 
than  the  center  of  the  spiral  arc,  the  divergence 
will  be  added  to  the  ordinate  of  the  circular  arc, 
and  for  one  farther  away  it  will  be  subtracted  from 
the  ordinate.  As  before,  the  degree  of  the  osculat- 
ing circular  curve  is  the  mean  of  the  Z>'s  at  the  end 
of  the  given  arc. 

Other  properties  may  be  found  by  ordinary  trig- 
onometric operations. 


20  SUMMARY  OF  PRINCIPLES 

SUMMARY  OF  PRINCIPLES 

27.  For  convenience  of  reference  the   principal 
formulas  will  be  repeated  here. 

D  =  a  L  and  L  =  — 

D\  =  a  L\  for  whole  spiral 

'  ..  '-*«*-*^=*?  ..............  1 

J  =  %  a  L\  =  Y*  Di  Li  for  whole  spiral  .  .  .  .  J 

j  =  .2910Z8  —  etc  ..............................  (6) 

*  =  100  Z,  —  .000762  a*  Lb  +  etc  .......  .  ......  (7) 


=  y$  A\—\aL\  for  whole  spiral 

z  ........  (10) 

(11) 
A'  ±<P  =  0'  +  0  +  lD'L  .....................  (12) 

o  =  .0727  aZi8  —  .  0727  DiL?  ...............  (13) 

Li=  3.71V  !JI  ................................  (14) 

(15) 


50  Zi  —  .000  127  a8  Zi6  .  .  .  .  (17) 


DEGREE  OP  CURVE  21 

Cff+0)  tan  %  I (18) 

}  exsec  %  1+  o (20) 

C=  100  L  —.00034  a*D> (21) 

u  =  x  —  y  cot  A "") 

I  (22) 
u  =  x  —  v  cos  ^ J 

v  =  -j£-j  =  ^-  L  +  .  000244  a*  U (23) 

An  inspection  of  the  formulas  and  demonstra- 
tions will  show  the  following  properties  of  the 
transition  spiral: 

28  Degree-of-curve. — The  degree-of-curve  at 
any  point  on  the  spiral  equals  the  degree  at  100 
feet  from  the  P.S.  multiplied  by  the  distance  along- 
the  spiral  from  the  P.S.  to  the  point  (Eq.  1).  This 
distance  must  be  expressed  in  units  of  100  feet 
(stations).  Thus,  if  a  =  2,  at  100  feet  from  the 
P.S.  the  spiral  will  be  a  2°  curve;  at  25  feet, 
(£=.25)  a  0°30r  curve;  at  450  feet,  (£  =  4.5)  a 
9°  curve.  ^£-  is  the  number  of  feet  of  spiral  in 
which  D  chang-es  one  degree.  Thus,  for  a  =  2  the 
spiral  increases  its  degree  of  curve  one  degree  for 
each  -ijfi-=50  feet;  for  0  =  f  one  degree  for  each 
=  150  feet. 


At  the  terminal  point,  the  P.C.C.,  where  the  spiral 
connects  with  the  main  curve,  D  will  sometimes 
be  represented  by  D\^  and  this  should  generally  equal 
the  degree  of  the  circular  curve  D$.  The  total 

length  of  the  spiral  will  be-^p*  If  a  —  2,  a  6°  degree 
would  require  a  spiral  3  stations  (300  feet)  long. 


22  SUMMARY  OF  PRINCIPLES 

29.  Angle  ^. — The  angle  ^  between  the  initial 
tangent  and  the  tangent  at  any  point  on  the  spiral 
(the  change  of  direction  corresponding  to  central 
angle  of  circular  curves)  (I/FP,  Fig.  1,  page  6)  in 
degrees  equals  (Eq.  2): 

(0)  One  half  of  a  times  the  square  of  the  distance 
in  100-ft.  stations  from  the  P.S.  to  the  point;  thus 
if  a  =  2,  for  300  ft.  from  P.S.,  Z=3,  and  J=  }4  X 
2X3*:=  9°.  Or 

(#)  One  half  of  the  product  of  this  distance  L  by 
the  degree-of-curve  of  the  spiral  at  the  given  point; 
thus  at  300  ft.  with  a  =  2,  D  =  6°,  and  J  =  %  X  3 
X6  =  9°.  Or 

(c)  One  half  of  the  square  of  degree-of-curve  at 
the  point  divided  by  a;  thus  at  300  ft.  with  a  =  2, 

J  =  #Xi'  =  9°. 

For  the  same  angle,  then,  the  spiral  is  twice  as 
long  as  a  circular  curve,  and  for  the  same  length 
the  angle  is  one  half  that  for  a  circular  curve  whose 
D  is  the  same  as  that  at  the  end  of  the  spiral. 

30.  Spiral  deflection  angle  #.—  The  spiral  deflec- 
tion angle  0  at  the  P.S.  from  the  initial  tangent  to 
any  point  on  the  spiral,  as  PAL  in  Fig.  1,  is  y§  A, 
or  i  a  L\      Thus,  for  a  point  300  ft.  from  the  P.S. 
(L=  3),  if  a=  2,  0  =  -fc  X  2  X  32=  3°.     If  the  result 
is  wanted  in  minutes,  since  -J-  X  60  =  10,  use  10  in- 
stead of  |.     For  105.4  ft.  with  a  =  2,  0  =  10  X  2  X 
(1.054)8  =  22'.     0  is  also  one  third  of  the  deflection 
angle  for  a  simple  curve  of  the  same  degree  as  the 
spiral  at  the  given  point.     Thus,  as  above,  the  de- 
flection angle  for  300  ft.  of  6°  curve  is  9°  and  0= y$ 
X9  — 3°. 

Tnese  values  are  subject  to  slight  corrections  for 


DEFLECTION    ANGLE  23 

^  larger  than  15°  or  20°  as  explained  in  the  deriva- 
tion of  the  formula  on  pag-e  10. 

31.  Tangent  at  point  on  spiral. — The  deflection 
ang-le  at  any  point  on  the  spiral  between  the  tan- 
g-ent at  this  point  and  the  chord  to  the  P.S.  (TLA 
in  Fig-.  1)  is  ^  —  6.     This  enables  the  tang-ent  to 
be  found.     For  A  less  than  15°,  the  value  f  J  or  2  Q 
is  sufficiently  accurate.     Thus,  for  the  preceding- 
example,  with  a  =  2,  for  the  point  300  ft.  from  the 
P.S.,  this  ang-le  is  2  0=  6°. 

32.  Deflection  angle  at  point  on  spiral. — For  de- 
flection angles  from  a  point  on  the  spiral  to  other 
points  on  the  spiral,  the  principle  that  the  spiral 
diverges  from  the  osculating-  circle  (circular  curve 
of  same  degree)  at  the  same  rate  that  the  spiral 
deflects  from  the  initial  tangent  is  of  service.     The 
ang-les  may  be  treated  in  three  ways,  as  follows: 

33.  Angles  from  tangent. — By  equation  (10)  the 
deflection  angle  between  the  tang-ent  at  a  transit 
point  on  the  spiral  and  the  chord  to  any  other  point 
on  the  spiral  (as  CBH,  Fig-.  4)  is  the  sum  or  differ- 
ence of  two  angles:     (1)  the  deflection  ang-le  for  a 
circular  curve  of  the  same  degree  as  the  spiral  at 
the  transit  point  for  a  leng-th  equal  to-  the  distance 
between  the  two  points,  and  (2)  the  spiral  deflec- 
tion angle  0  for  a  leng-th  of  spiral  equal  to  the  dis- 
tance between  the  two  points.     The  latter  ang-le  is 
plus  if  the  desired  point  is  further  from  the  P.S., 
and  minus  if  nearer,  than  the  point  from  which  the 
deflections  are  made. 

Thus,  if  a  =  2  and  the  transit  be  at  B  (Fig-.  4), 
250  ft.  from  the  P.S.,  the  degree-of- curve  at  the 
transit  point  will  be  5°,  and  the  deflection  ang-le 


24  SUMMARY  OF  PRINCIPLES 

CBH  to  set  a  point  150  ft.  ahead  will  be  the  sum  of 
3°  45',  (#  of  150  ft.  of  5°  curve)  and  45',  (the 

2 

spiral  deflection  angle  for  150  feet,  10  X  2  X  1  •  5) 
or  4°  30'.  For  D,  150  ft.  back,  it  would  be  3°  45' 
—  45'  =  3°0'. 

34.  Angles  from  chord. — Likewise  by  equation 
(11)  the  angle  CBE,  Fig-.  4,  (deflection  angle  from 
chord  to  P.S.,)  may  be  calculated  by  adding  the 
spiral  deflection  angle  0  for  the  point  C  (GAC)  to  % 
the  product  of  the  degree-of-curve  at  B  by  the  num- 
ber of  stations  from  the  P.S.  to  C.  For  a  =  2  and 
the  transit  at  B,  250  ft.  from  the  P.S.,  the  degree- 
of-curve  at  the  transit  point  is  5°,  and  the  angle 
CBE  to  locate  the  point  C  150  ft.  ahead  and  400  ft. 
from  the  P.S.,  will  be  (i  X  2  X  48  =  5°  20')  +  (i  X  5 
X4=3°  20')=  8°  40'.  For  the  point  D  100  ft. 


FIG.  4. 

from  the  P.S.,  the  angle  DBA  will  be  (-J-  X  2  X  I2  = 
20')  +  (|X  5X  1  =  50')  =  1°  10'.  This  method  is 
applicable  whether  the  point  to  be  located  is  nearer 
to,  or  farther  from,  the  P.S.  than  the  transit  point. 
It  permits  the  calculation  of  the  spiral  deflection 
angles  at  P.S,  for  the  whole  spiral  and  the  deter- 
mination of  the  angles  between  the  chords  in  ques- 


ABSCISSA  25 

tion  by  adding*  to  these  spiral  deflection  angles  the 
angles  -J-  Df  L,  where  D'  is  the  degree-of -curve  at 
the  transit  point  and  L  is  the  distance  from  P.S.  to 
the  point  to  be  located. 

35.  Angles  with  initial   tangent. —Equation  (12) 
gives  the  deflection  angles  from  a  line  parallel  with 
the  initial  tangent.     The  results  are  the  same  as  if 
the  spiral  deflection  angle  0'   for  the  transit  point 
were  added  to  those  from  the  chord  found  in  the 
preceding  paragraph. 

36.  Divergence    from    osculating    circle.— The 
spiral  diverges  from  its  osculating  circle  (circular 
curve  of  the  same  degree)  at  any  point  at  the  same 
rate  that  the  spiral  deflects  from  the  initial  tangent, 
and  the  distance  between  the  circle  and  spiral  is  the 
same  as  thejy  for  an  equal  length  of  spiral. 

This  enables  the  spiral  to  be  located  by  offsets 
measured  from  the  circular  curve.  By  this  method 
half  of  the  spiral  may  be  located  from  the  initial 
tangent  and  half  from  the  produced  circular  curve, 
the  offsets  for  the  two  being  the  same  for  the  same 
distances  from  the  P.S.  and  the  P.C.C.  respectively. 
See  Fig.  5. 

37.  Abscissa  oc. — The  distance  in  feet  along  the 
initial  tangent  to  the  perpendicular  to  a  point  on 
the  spiral  is  equal  to  the  length  along  the  spiral  in 
feet  less  the  quantity  .000762  a2  U>,  where  L  is  the 
length  along  the  spiral  expressed  in  units  of  100  ft. 
A  convenient  way  to  find  x  is  to  have  this  quantity 
tabulated  for  given  spirals  as  a  correction,  or  it  may 
easily  be  found  from  tabulated  values  of  such  a  cor- 
rection for  a  =  1  by  multiplying  by  a2.  For  extreme 
lengths  another  term  may  be  needed.     As  an  illus- 


26  SUMMARY  OF  PRINCIPLES 

tration,  with  a=l  for  200  ft.  the  correction  to  be 
subtracted  from  200  ft.  to  find  x  is  .000762  X  32  = 
.02  ft.,  a  small  quantity. 

38.  Ordinate  y.  —  The  ordinate  y  (perpendicular 
distance  from  the  initial  tangent  to  a  point  on  the 
curve)  in  feet  equals  .291  times  the  product  of  a  by 
the  cube  of  the  distance  along-  the  spiral  from  the 
P.S.  to  the  point  expressed  in  units  of  100  ft.   (sta- 
tions).    It  therefore  varies  as  the  cube  of  the  dis- 
tance from  P.S.     Knowing  y  for  one  point,  the  y 
for  a  second  point  may  be  computed  from  it  by  this 
relation.     D  L?  may  be  substituted  for  a  Z3.     For 
extreme  lengths,  a  third  term  may  have  to  be  con- 
sidered.    As  an  illustration,  with  a  =  2  for  200  ft. 
(£=2),^  =  .291X8  =  2.33  ft.     For  100  ft. , y  is  one 
eighth  as  great;  for  400  ft.  y  may  be  used  as  eight 
times  as  great,  though  the  use  of  the  next  term  of 
the  series  would  change  this  somewhat. 

39.  Offset  o. — The  offset  o   between  the  initial 
tangent  and  the  parallel  tangent  from  the   main 
curve   produced   backward,  (BD,   Fig.   1),    in   feet 
equals  .0727  times  the  product  of  a  by  the  cube  of 
the  length  of  the  whole  spiral  in  stations,  or  .0727 
times  the  square  of  the  length  of  spiral  and  the 
degree  of  main  curve.     This  ordinate  is  approxi- 
mately one  fourth  of  the  ordinate  y  of  the  end  of 
spiral.     The  spiral  bisects  the  offset  at  a  point  half- 
way between  the  P.S.,  and  the  P.C.C.     (Eq.  11.) 
BE  =  ED.     AE=  EL.     (Fig.  1.)    The  slight  error 
in  this  is  discussed  in  the  derivation  of  the  formu- 
las (page  16) .     The  value  of  o  may  best  be  discussed 
by  means  of  one  of  the  tables. 

40.  Calculation  from  known  values.— When  the 


TANGENT   DISTANCES  27 

length  of  the  spiral  is  not  so  great  that  a  second  or 
correction  term  is  needed  for  the  values  of  0,  y,  ^, 
etc.,  it  is  seen  from  equations  (9),  (6),  (2),  etc., 
that  these  functions  vary  as  the  square  and  cube  of 
the  distance  L  and  may  be  calculated  from  any 
known  value.  Thus  if  e  for  400  ft.  is  2°  40',  for 


300ft.,  «=*!=-      ~     X(2°40')=l°30'.Ify 


for  400  ft.isl8.S9,for300ft.iy=-3<yl=  j  X  18.59 

=  7.85.  The  deflection  ang-le  varies  as  the  square 
of  the  distance  and  the  ordinate  as  the  cube  of  the 
distance  from  the  P.S. 

41.  /and   C.—  The   distance  t  from  the  P.S.   to 
this  offset  (  AB,  Fig-.  1)  is  found  by  subtracting-  the 
correction  .000127  a2  Li5  from  the  half  leng-th  of  the 
curve  in  feet.     (Eq.  17.)     Generally  this  correction 
term  is  quite  small.      As  stated  on  pag-e  15  this 
term  may  be  tabulated,  and  it  may  also  be  obtained 
for  a  given  length  of  spiral  by  multiplying  tabulated 
values  for  a=  \  by  the  square  of  the  a  of  the  given 
spiral.     For  this  use  see  pages  28  and  30. 

The  long  chord  C  is  found  by  subtracting  the  cor- 
rection, .000338  a*  U>  from  the  length  of  the  curve 
in  feet.  (Eq.  21.)  This  correction  may  be  found 
by  multiplying  the  oc  correction  for  the  same  length 
of  spiral  by  four  ninths. 

42.  u  and  v.  —  The  spiral  tangent-distances  u  and 
v  (AT  and  TL,  Fig.  1)  are  found  by  equations  (22) 
and  (23).     v  can  be  found  most  easily  by  taking 
one    third    of    the  tabulated  values  of  the  x  cor- 
rection and  oubiracting  thia  froM|  one   third  of  the 
length  of  the  spiral  in  feet. 

adding  this  tc 


28  THE  TABLES 

'    THE  TABLES 

43.  The  computations  may  be  shortened  by  the 
use  of  the  tables. 

Tables  I-XI  give  the  values  of  the  principal  parts 
of  the  transition  spiral  for  the  following-  values  of 
*  :  r<  i  t,  1.  l#i  1^>  2,  2#,  8#,  5  and  10.  The 
column  headed  "Leng-th"  is  the  distance  in  feet 
along-  the  spiral  from  the  P.S.  to  any  point  on  the 
spiral,  and  is  equal  to  100  times  the  L  of  the  formu- 
las. The  column  headed  "x  COR."  gives  the  cor- 
rection to  be  subtracted  from  this  distance  in  feet 
along-  the  spiral  to  obtain  #,  and  that  headed 
"/  COR."  gives  the  correction  to  be  subtracted 
from  the  half  leng-th  of  the  spiral  in  feet  to  obtain  /. 
Both  /  COR.  and  o  are  to  be  taken  from  the  line  for  the 
full  leng-th  of  the  spiral.  For  example,  by  Table 
IV,  with  a  =  1,  to  connect  with  a  5°  curve,  the 
leng-th  of  spiral  is  500  ft.  and  L  =  5  ;  the  chang-e  of 
direction  Ji  is  12°  30' ;  the  offset  o  to  P.C.  of  circular 
curve  is  9.07  ft.;  /  is  250— .4=249.6  ft.;  x  is  500  — 
2.37=497.63  ft.;  and  the  values  of  D,  J,  9,  y  and  oc 
COR.  for  points  200,  210,  220  ft.,  etc.,  distant  from 
the  P.S.  are  found  in  the  line  with  200,  210,  etc. 

To  find  the  long-  chord  to  P.  S.,  C,  subtract  .445  of 
x  COR.  from  the  leng-th  of  the  curve  in  feet.  To 
find  the  spiral  tang-ent-distance,  z>,  add  one  third  of 
x  COR.  to  one  third  of  the  leng-th  of  the  spiral  in 
feet. 

Tables  I  -  IV  have  the  values  of  4  and  9  calcu- 
lated to  the  nearest  tenth  of  a  minute,  and  Tables 
V-VII  to  the  nearest  half  minute.  While  this  pre- 
cision is  not  usually  necessary,  it  may  be  of  service 
where  the  sum  of  two  or  more  angies  is  used. 


INTERPOLATION 


44.  Interpolation.  —  To  find  values  intermediate 
between  the  distances  given  in  the  tables,  interpo- 
late by  multiplying-  one  tenth  of  the  difference  be- 
tween consecutive  values  by  the  number  of  addi- 
tional units.  Thus  Table  IV  gives  A  for  400  ft. 
as  8°00';  for  410  ft.,  8°24'.3.  One  tenth  of  the  dif- 
ference between  these  is  2  '.4.  For  406.8  ft.,  add 
6.8  X  2.4=  16'.  3  to  8°00',  giving  8°16'.  Fory,  add 
6.8  X  .143  =  .97  to  18.59,  giving  19.56  ft.  For  0, 
add  6.8  times  one  tenth  of  .36  to  4.65  giving  4.89. 
D  is  4.  068  or  4°  4'.  08. 

Interpolation  may  also  be  made  for  other  columns. 
Thus  if  o  is  given  as  7.0  ft.  and  a  =  1%,  by  Table 
V  the  length  of  spiral  will  be  between  420  and  430 
ft.  Interpolating,  as  o  increases  0.5  ft.  in  10  ft.  of 
length,  the  .28  will  be  gained  in  5.6  ft.  and  the 
length  is  425.6  ft.  Interpolation  for  ^,  D,  etc., 
may  then  be  made  as  before.  Again  ,  for  D  =  4°  16  '  , 
still  using  0=1#,  the  length  is  between  340  and 
350  and  is  ^X  10  —  1.33  ft.  more  than  340,  making 
341.33. 

In  general  this  interpolation  gives  accurate  re- 
sults and  no  correction  need  be  made.  For  A  the 
error  in  interpolation  with  values  of  a  greater  than 
5  may  need  to  be  taken  into  account.  To  find  ex- 
act values  of  ^,  deduct  from  the  interpolated  values 
a  times  the  following  quantities  :  For  a  length  in 
feet  ending  with  1,  .027';  2,  .048'  ;  3,  .063'  ;  4,  .072'  ; 
5,  .075';  6,  .072';  7,  .063'  ;  8,  .048'  ;  9,  .027'  .  It  can 
easily  be  determined  whether  this  correction  need 
be  considered.  The  difference  arises  from  the  fact 
that  the  square  of  numbers  does  not  increase  uni- 
formly. For  the  other  columns  the  errors  of  inter- 


30  .  THE  TABLES 

polation  are    very   slight   and   may   be   neglected. 

45.  General  use  of  Table  IV.— Table  IV  has  been 
carried  to  several  decimal  places  to  permit  its  use 
for  values  of  a  other  than  1.  To  calculate  values 
for  another  a,  multiply  the  tabular  value  of  D,  A,  9, 
o,  or y  in  Table  IV  for  the  distance  from  the  P.S. 
to  the  point  on  the  spiral  by  the  a  of  the  spiral  used, 
and  the  x  COR.  and  /  COR.  by  the  square  of  the  a  of 
the  spiral.  Thus  if  a  —  2.2  and  L=  3.1,  multiply 
the  D,  J,  #,  o,  and  y  opposite  310  by  2.2,  and  the  x 
COR.  and  t  COR.  by  the  square  of  2.2.  The  values 
of  jy,  o,  and -x  COR.  obtained  in  this  way  are  subject 
to  slight  errors  for  large  values  of  a  if  J  is  more 
than  18°,  but  fortunately  y  for  a  distance  greater 
than  half  of  the  length  of,  the  spiral  is  seldom 
needed,  and  as  the  error  of  this  and  the  errors  in  o 
and  x  COR.  are  ordinarily  small  the  correction  may 
generally  be  neglected.  The  amount  of  this  error 
may  be  found  by  the  method  given  in  a  succeeding 
paragraph.  The  error  in  0  is  discussed  on  page  10. 

To  use  Table  IV  for  another  a,  it  may  be  desira- 
ble first  to  determine  the  length  of  the  spiral  by 
dividing  the  D\  of  the  required  spiral  by  a  or  to 
determine  it  from  o.  Thus,  for  a  =  1.5,  to  connect 
with  a  6°  curve,  divide  6  by  1.5,  which  gives  L\  =  4; 
that  is,  the  whole  spiral  will  be  400  ft.  long,  and 
the  properties  for  the  spiral  may  be  computed  by 
multiplying  those  in  the  line  with  the  required  dis- 
tance by  1.5.  In  other  words  it  must  be  borne  in 
mind  that  the  distances  in  the  column  of  lengths 
remain  unchanged  with  new  values  of  a,  and  the 
quantities  in  all  the  other  columns  will  be  changed 
for  a  other  than  1. 


CORRECTIONS 

46.  Corrections  for  calculations. — For  the  cal- 
culation of  tables  and  other  work  requiring*  the 
recognition  of  a  further  term  in  the  equations,  the 
value  of  the  second  term  of  the  o  series  (.0000002 
as  U;  eq.  (13)  )  and  of  the  second  term  of  the  y 
series  (.  00000158  a*  L\  eq.  (6)  )  may  be  obtained 
by  multiplying-  the  quantities  in  the  following- 
table  by  #3;  and  the  third  term  of  the  x  series 
(.00000000268  a*  L\  eq.  (7)  )  by  «*-.  These  terms 
for  o  and  y  are  neg-ative,  and  the  term  for  x  is  to  be 
subtracted  from  the  x  COR. 


L 

0 

y 

2.50 

.0010 

3.00 

.0004 

.0035 

3.50 

.0013 

.010 

4.00 

.0032 

.026 

4.50 

.0074 

.059 

5.00 

.015 

.124 

5.50 

.030 

.241 

6.00 

.055 

.442 

6.50 

.097 

.775 

.0007        '; 

.002 
.005 
.012 
.027 
.055 
7.00  .163  1.301  .108 

For  making-  corrections  on  results  obtained  from 
Table  IV  for  a  other  than  1,  subtract  from  the 
product  of  the  multiplication  used  to  obtain  the 
desired  distance  a  (a2  —  1)  times  the  value  from  the 
above  table  in  obtaining-  o  and  j,  and  a  (a3  —  1)  times 
the  value  from  the  table  in  obtaining-  the  x  COR. 

47.  Table  of  ordinates.— By  Table  XII  the  ordi- 
nate  from  the  tang-ent  or  from  the  circular  curve  at 
a  decimal  part  of  the  half  leng-th  of  the  spiral  may 
be  obtained  by  the  multiplication  of  o  of  the  spiral 


32  CHOICE  OF  SPIRAL 

by  the  factor  given  in  the  table.  See  method  by 
co-ordinates  and  Fig-  5.  It  should  not  be  forgotten 
that  Tables  I  -  X  give  ordinates,  and  that  values 
for  intermediate  points  may  easily  be  interpolated. 

48.  Table  of  offsets. — Table  XX  gives  values  of 
o  and  L  for  various  values  of  a.  Within  reasonable 
limits  it  will  bear  interpolation,  both  for  interme- 
diate values  of  a  and  D  and  to  determine  a  for  in- 
termediate values  of  o.  It  is  of  service  in  location 
problems. 

Tables  XIII  and  XIV  are  described  under  Uni- 
form Chord  Leng-th  Method.  The  tables  for  street 
railway  curves  are  described  under  Street  Railway 
Spirals. 


CHOICE  OF  a  AND  LENGTH  OF  SPIRAL 

49.  The  selection  of  a  and  with  it  the  leng-th  of 
spiral   require  consideration.     The  value  of  a  to  be 
used  is  dependent  upon  the  speed   of   trains,  the 
maximum  degree-of-curve,  the  leng-th  of  tang-ents, 
the  permissible  offset  of  the  line  for  the  topograph- 
ical conditions  in  question,  the  distance  in  which 
the  superelevation  of  the  outer  rail  may  be  attained, 
etc.,  and  hence  is  subject  to  a  wide  range  of  condi- 
tions.    It  may,  however,  aid  the  engineer's  judg-- 
ment  to  discuss  these  conditions  briefly. 

50.  Effect  of  speed. — For  the  same  rolling-  stock 
and  for  the  same  comfort  in  riding-,  it  would  seem 
that   a   given   amount   of   superelevation   must  be 
attained  in  the  same  leng-th  of  time;  and  hence  it 
is  probable  that  a  should  vary  nearly  inversely  as 
the  cube  of  the  speed  of  train.     This  conclusion  also 


ATTAINMENT  OF  SUPERELEVATION  33 

emphasizes  the  desirability  of  spiraling  curves  used 
under  high  speeds. 

Assuming*  that  a  =  1  is  a  proper  value  for  speeds 
of  50  miles  per  hour,  this  principle  would  suggest 
the  following  maximum  values  of  a:  60  miles  per 
hour,  y<z\  50  miles  per  hour,  1;  40  miles  per  hour,  2; 
30  miles  per  hour,  3j4;  25  miles  per  hour,  5;  20 
miles  per  hour,  10.  While  for  the  very  high  speeds 
this  may  seem  to  require  unnecessarily  long  spirals 
and  for  low  speeds  short  spirals,  yet  a  =  1  has  given 
satisfactory  results  at  speeds  of  50  to  60  miles  an 
hour,  and  a  =  2  at  40  to  50  miles  an  hour,  and  for 
60  miles  an  hour,  a  =  YZ  is  not  too  small.  Of 
course,  in  any  case,  longer  spirals  and  smaller 
values  of  a  will  give  smoother  riding  curves. 

51.  The  speed  of  trains  may  be  limited  by  the 
maximum  superelevation  allowable  on  the  sharper 
curves.     Under  usual  practice  the  requirement  of 
maximum  superelevation  would  limit  the  maximum 
degree-of-curve  for  speeds  of  60  miles  an  hour  to  3°, 
for  50  miles  to  4°,  for  40  miles  to  6°,   for  30  miles 
to  12°,  etc.     Where  the  track  is  not  used  for  slow 
trains  and  a  superelevation  of  more  than  7  or  8 
inches  is  allowable,    somewhat  higher   speeds   on 
such  curves  may  be  used.     The  maximum  speed  of 
train,  however,  will  be  the  governing  consideration 
in    the    choice    of    a    rather  than   the    maximum 
degree-of-curve. 

52.  Attainment  of  superelevation. — The  rate   of 
attaining  the  superelevation  is  sometimes  given  as 
the  governing  consideration,  but  in  reality  this  rate 
is  governed  by  the  speed.      The  distance  in  which 
the  outer  rail  should  attain  an  elevation  of  1  inch 


34  CHOICE  OF  SPIRAL 

will  not  be  the  same  for  a  speed  of  60  miles  an  hour 
as  for  one  of  40  miles.  The  schedule  of  maximum 
values  of  a  for  various  speeds  as  given  above  in- 
volves, approximately,  attaining*  1  inch  of  elevation 
in  the  following-  distances  :  60  miles,  80  feet;  50 
miles,  53  feet ;  40  miles,  44  feet;  25  miles,  40  feet. 
Speed  and  amount  of  superelevation  should  g"ov- 
ern  the  lenglh  of  spiral,  and  rate  of  attainment  is 
subordinate. 

53.  It  may  be  convenient  for  maintenance-of-way 
work  to  arrang-e  the  spiral  so  that  the  supereleva- 
tion  is   attained  at  a  definite  rate  per  100  ft.    of 
length  of  spiral.     Let  k  be  this  rate,  expressed  in 
inches  of  superelevation  attained  in  100  feet.     Let 
h   be  the   superelevation   in  inches  per  degree   of 
curve ;  for  a  3°  curve,  3^,  etc.     Then 

i         i.       D  i  k 

k  =  a/i  =  -r-  /i.  a  =  -7-. 

L  h 

54.  The  following-  table  shows  the  values  of  #, 
which  give  rates  of  1,  1^    and  2  inches  of  super- 
elevation  attained  in   100  ft.    for   the    amount   of 
superelevation   per  degree  of   curve   given  at   the 
head  of  the  columns. 

VALUES  OF  a 
Elevation  per  degree    ^     Y^     1     1#     1^     2     2^ 

a  for  k  equal  to  1  in.  per  100  ft.      2  1|  1  I  §  £            i 
a  for  Tc  equal  to  1£  in.  per  100  f t.    3  2  1|  f  1  i             I 
a  for  k  equal  to  2  in.  per  100  ft.    4  2|  2  II  1J  1              f 
Velocity  in  miles  an  hour  cor- 
responding- to  superelevation  26.9  33.0  38.1  42.6  46.6  53.8  60.2 

The  length  of  spiral  for  a  =  %  is  200  ft.  for  each 
degree  of  curve  ;  for  a  =  f,  125  ft.;  for  a  —  f,  150  ft., 
.etc.  When  tables  are  not  given  for  the  a  used,  the 
values  may  be  calculated  from  the  tables  by  multi- 


MINIMUM  SPIRALS  35 

plication  or  other  process.  Thus,  for  a  —  1^,  double 
the  values  for  a=\\  for  a  =  f  multiply  those  from 
a  =  %  by  1£  or  those  from  a  =  1  by  f . 

55.  The  amount  of   superelevation  per  degree  of 
curve  here  used  is  calculated  from  .00069  V'1,  where  V 
is  the  velocity  in  miles  per  hour.     This  gives  the 
number  of  inches  per  degree  to  counteract  the  cen- 
trifug-al  force,  and  is  based  on  distance  from  center 
to  center   of  rail.      This  amount  is  used  here  be- 
cause it  is  the  most  common  value  ;  for  other  super- 
elevations comparisons  may  readily  be  made  with 
the  figures  here  given.     For  convenience  in  mainte- 
nance-of-way  work  it  may  be  desirable  to  establish 
the  superelevation  at  a  convenient  amount  near  the 
value  calculated  for  the  assumed  velocity,  an  allow- 
able practice  since  the  assumed  velocity  may  not  be 
realized.     Thus  2  inches  per  degree  may  be  used  in 
place  of  lg,  etc. 

56.  Minimum  spirals.— For  a  given  value  of  a 
there  may  be  a  question  as  to  how  flat  a  curve  may 
profitably  be  spiraled,     The  spiral  should  certainly 
vary  enoug*h  from  the  position  of  a  simple  circular 
curve  that  the  distinction  may  not  be  obliterated  by 
the  inaccuracies  of  track  work;  otherwise  it  will  be 
as  advantageous  to  beg-in  the  superelevation  an  equal 
distance  back  on  the  tang-ent.     The   limits  given 
in   a   iortner  edition  have  been  criticised  by  engi- 
neers of   maintenance  of  way,    and   experience  on 
prominent  roads  indicates  that  the  minimum  limit 
of  o  there  set,  0.6  to  1  ft.,  was  too  hig-h.     It  seems 
that  the  gradual  change  of  direction  between  the 
trucks  and  the  car  body,  and  the  fitting  of  elevation 
to  the  curvature  by  the  spiral,  are  advantageous  to 


36  CHOICE  OF  SPIRAL 

smooth  riding-  even  when  the  change  in  alignment 
is  slight.  Experience  seems  to  indicate  that 
for  a  =  )4  curves  above  30 '  may  be  spiraled;  for 
0  =  1,  1°  and  above;  for  a  =  2,  2° ;  for  a  =  31/! ,  3° ; 
for  a  =  5,  4°;  for  a  =  W,  6°.  For  curves  lighter 
than  these  any  advantag-e  seemingly  found  by 
spiraling  would  probably  be  obtained  by  beginning 
the  superelevation  back  on  the  tangent. 

In  any  case  decreasing  the  value  of  a  and  thus 
increasing  the  length  of  the  spiral  will  increase  the 
efficiency  of  the  spiral  and  better  the  riding  quali- 
ties of  the  curve.  This  view  needs  emphasizing, 
and  too  much  should  not  be  expected  of  short  spirals. 

57  Selection  of  a  and  length  of  spiral. — The  se- 
lection of  #,  then,  must  be  a  matter  to  be  left  to  the 
judgment  of  the  engineer.  As  a  guide  the  follow- 
ing table  containing-  values  of  a  which  have  given 
satisfactory  results  at  the  speeds  noted  is  given. 
Lower  values  of  a  are  of  course  advantag-eous;  as, 
for  example,  at  a  speed  of  40  miles  an  hour  a  =  \l/s 
or  even  1  will  make  a  more  efficient  easement  than 
the  one  given.  Higher  values  of  a — shorter  spirals 
—  may  be  necessary  in  many  cases,  but  it  must  be 
understood  that  they  will  not  be  so  satisfactory. 
The  column  headed  * 'Minimum  curve  spiraled"  is 
the  lightest  curve  which  it  is  considered  desirable 
to  spiral  with  the  value  of  a  given  opposite.  "Maxi- 
mum curve1'  is  fixed  at  the  given  speed  by  the  limit 
of  superelevation;  at  lower  speeds  this  a  may  profit- 
ably be  used  for  sharper  curves.  The  speeds  are 
given  in  miles  per  hour  and  the  elevations  in  inches. 


P.S.,  P.C.C.,   AND    P.C.  37 

MINIMUM  SPIRAL  FOR  MAXIMUM  SPEED 


Maximum 
Speed. 

a 

Maximum 
Curve. 

Min.  Curve 
Spiraled. 

I^eng-th  per 
Degree. 

Elev.  per 
Degree. 

60 

YZ 

3° 

30' 

200 

2.y2 

50 

1 

4° 

1° 

100 

ift 

40 

2 

7° 

2° 

50 

i# 

30 

3*4 

11° 

3° 

30 

& 

25 

5 

14° 

4° 

20 

y* 

20 

10 

25° 

5° 

10 

T2A 

For  shorter  spirals,  the  following-  values  of  a  are 
consistent  with  each  other:  60  miles  per  hour,  1; 
50  miles  per  hour,  1^;  40  miles  per  hour,  3^;  30 
miles  per  hour,  67^3;  25  miles  per  hour,  10. 

LOCATION  OF  P.S.,  P.C.C.,  AND  P.C. 

58.  Location  from  intersection  of  tangents.— 
When  the  tangents  have  been  run  to  an  intersection, 
the  P.S.  (A,  Fig-,  i,  page  6)  may  be  located  by 
measuring-  back  on  the  tang-ent  from  the  point  of 
intersection  a  distance  equal  to  the  tang-ential 
distance  T  (equation  18) .  This  distance  may  also 
be  computed  by  adding-  t+o  tan  YZ  /to  the  tang-en- 
tial distance  of  the  circular  curve  as  ordinarily 
calculated.  (See  section  21).  The  P.C.  (D,  Fig-. 
i)  may  be  located  by  calculating-  o  and  offsetting 
this  amount  at  a  point  on  the  tangent  distant  /  from 
the  P.S.  (See  section  19.)  /  may  be  found  by 
subtracting  the  /  cor.  of  the  tables  from  the  half 
length  of  curve  in  feet.  Thus,  for  400  ft.  of  spiral 
with  a  •  =  2,  by  Table  VII  /  cor.  is  .5  ft.  and  /  =  200 
—0.5  =  199.  5  ft.  The  P.C.C.  (L,  Fig.  i)  may  then 


38  P.S.,   P.C.C.,   AND    P.C. 

be  located  by  running- the  spiral  from  the  P.S.,  or 
by  locating-  the  circular  curve  from  the  P.C.  for  a 
distance  >^  L\. 

59.  Location  from  P.C   of  a  curve  without  spiral. 
In  case  a  simple  curve  has  been  run  without  pro- 
vision for  a  spiral   and  without  offsets,    that  is   in 
the  usual  way,  it  will  be  necessary  to  change  the 
position  of  the  circular  curve.     The  distance  of  the 
P.S.  back  of  the  P  C.  of  the  old   simple  curve   will 
be  t-\-o  tan  ^  /,  /being-  the  total  intersection  ang-le. 
The  new  curve  will  come  inside  the  old  but  will  not 
be  exactly  parallel  to  it. 

60.  Location  from  P.C.  of  offsetted  curve. — If  a 

simple  curve  has  been  run  for  use  with  spiral,  as 
DLH  in  Fig-.  1,  page  6,  o  may  be  computed,  the 
offset  measured  to  B  and  the  distance  t  (AB)  meas- 
ured to  locate  the  P.S.  (A).  The  length  >4  Zi  meas- 
ured from  the  P.C.  on  the  circular  curve  will  locate 
the  P.C.C.  (L).  Similarly  if  the  tangent  is  fixed, 
the  curve  may  be  located  by  first  making-  the  offset 
from  the  tangent  to  the  P.C. 

61.  If  both  P.C.  and  tangent  are  fixed  with  an 
offset  0=BD  between  them,  a  may  be  found  from 
a=.269/v/I?[/  or  a  and  L  may  be  found  from  Table 

0 

XX.  After  finding /,  the  P.S.  may  be  located  in 
the  usual  manner.  For  a  5°  curve  with  o  =  W  ft., 
by  equations  (14)  and  (16)  Z  =  525.3  ft.  and  a  = 
.952.  Since  with  a  =  1  and  this  length  of  spiral  t 
cor.  =.5,  the  correction  to  be  used  here  is  0.5X«2, 
and  /  =  262.65— .45  =  262.2  ft.  This  method  is  a 
great  convenience  where  it  is  desired  on  account  of 
the  ground  to  throw  the  curve  in  or  out  without 


LOCATION   BY    CO-ORDINATES.  39 

changing  the  tangent,  or  where  a  similar  change 
in  the  tangent  is  desired  without  a  change  in  the 
curve,  the  connection  to  be  made  by  means  of  a 
suitable  spiral. 

LAYING  OUT  THE  SPIRAL  BY  CO- 
ORDINATES. 

62.  With  the  initial  tang-ent  as  axis  of  X  and 
the  P.S.    as   the   origin  of   co-ordinates,  it  is   not 
difficult  to  locate  points  on  the  spiral  by   means   of 
co-ordinates.       These    may     be     calculated     from 
equations  (6)  and  (7),  or  they  may  be  taken  from 
the  tables.      Beyond  B  of  Fig-.  1,  pag-e  6,  the   or- 
dinates  become  larg-e  and  the  x  correction  may  be 
considerable.     For  long-  spirals,  the  second  term  of 
the   y  series,    may    need   to   be    considered.     The 
property  that  the  spiral  diverg-es  from  the  circular 
curve  at  the  same  rate  as  from  the  initial  tang-ent 
is  of  service.     Between  E  and  the  P.C.C.  (L)  meas- 
ure the  ordinate  or  offset  from  the  circular  curve, 
using-   for  this  offset  at  a  point  a  given  distance 
from  the  P.C.C  the  ordinate  y  of  the  spiral  from 
the  tangent  at  the  same   distance   from   the   P.S. 
Thus,  for  0  =  1,  by  Table  I,  for  a  point  200  ft.  from 
the    P.S.,  jy-2.33    ft.     To  locate  a  point  on  the 
spiral  200  ft.  from  the  P.C.C.,  offset  from  the  circu- 
lar curve  this  same  distance,  2.33  ft. 

63.  Knowing  the  y  for  any  point  on  the   curve, 
the  y  for  any  other  point  within  ordinary   limits 
may  be  found  by  multiplying  the  former  number  by 
the  cube  of  the  ratio  of  the  distances  from  the  P.S. 
to  the  respective  points  ;  thus,  if  for  a  point  250  ft. 
from    the   P.S.,  jy  =  4.54   ft.,  y  at    300    ft.   equals 


4O  LOCATION   BY   CO-ORDINATES. 

(lf)3X4.54— 7.85  ft.  Similarly,  points  may  be  lo- 
cated by  offsets  from  the  circular  curve,  the 
distance  being  measured  from  the  P.C.C. 

64.  If  the  length  of  the  half  spiral  be  divided 
into  an  integral  number  of  parts  (See  Fig.  5),  any 
ordinate  from  the  tangent  or  from  the  circular 
curve  may  easily  be  calculated  from  o,  by  multiply- 
ing o  by  one  half  the  cube  of  the  ratio  of  the  num- 
ber of  parts  this  point  is  from  the  P.S.  to  the 
whole  number  of  parts.  The  following  table  gives 
the  factor  by  which  the  o  of  the  spiral  may  be  mul- 


RS.        2 


FIG.  5. 


RC.C. 


tiplied  to  determine  the  y  of  the  point  when  the 
length  of  the  half  spiral  is  divided  into  10  parts. 


TABLE    OF    FACTORS    FOR    ORDINATKS. 
To  find  yt  multiply  o  by  the  factor. 


Ratio  to  half 
length. 

O.I 

0.2      0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

I.O 

Factor. 

.0005 

.004J  .014 

.032 

.063 

.108 

.172 

.256 

.365 

.500 

As    an   example,    with   a  —  i,  o  for   a    5°    curve 
(spiral  500  feet  long)  is  9.07  ft.     The  half  length 


LOCATION    FROM    SPIRAL   TANGENTS.  41 

of  the  spiral  is  250  ft.  and  one  tenth  of  this 
distance  is  25  ft.  y  at  100  ft.  (0.4  of  the  half 
length)  is  .032X9.07=2.9  ft.  Similarly,  2.9  ft. 
will  be  the  offset  from  the  circular  curve  at  a  point 
100  ft.  from  theP.C.C. 

To  divide  the  half-length  of  spiral  into  5  parts 
and  the  whole  spiral  into  10  parts,  use  the  even 
numbered  tenths  of  the  above  table. 

For  intermediate  values,  interpolation  in  the 
above  table  will  give  reasonably  accurate  results. 
This  enables  interpolation  for  quarter  points,  and 
other  fractional  parts  ;  thus,  for  .67  of  half  length 
the  factor  is  .153. 

The  results  by  the  above  table  are  subject  to 
error  in  the  hundredths  place,  but  for  usual  cases 
are  within  .02  ft. 

65.  Still  another  method  is  to  measure  ordi- 
nates  from  the  initial  tangent  for  about  two  thirds 
of  the  length  of  the  spiral  and  for  the  remaining 
distance  to  measure  ordinates  or  offsets  from  the 
terminal  spiral  tangent  (TL,  Fig.  1,  page  6). 
The  offsets  from  the  terminal  spiral  tangent  will 
be  the  difference  between  the  offset  for  the  pro- 
duced circular  curve  DL  and  the  y  for  a  spiral, 
both  for  a  length  equal  to  the  distance  from  P.C.C. 
to  the  point  to  be  located.  As  this  distance  will  be 
less  than  one  third  the  spiral  length,  the  approxi- 
mate formula  for  tangent  offset,  .873  D\  U,  may 
ordinarily  be  used.  Di  is  the  degree  of  the  spiral  at 
the  P.C  C.  and  L  here  will  be  used  as  the  distance 
from  the  PC.C.  to  the  desired  point.  The  offset 
will  then  be  .873  Di  U— .291  a  U.  While  the  offsets 
are  longer  than  those  from  the  circular  curve,  the 


42  LOCATION  BY  TRANSIT   ANGLES 

measurements  are  made  from  the  tangent  and  the 
circular  curve  need  not  be  run.  As  an  example 
take  a  -  1,  and  D\  =  4°.  Length  of  spiral  is  then 
400  ft.  For  a  point  50  ft.  back  of  P.C.C.  (L  -  .5) 
the  offset  is  .87— .04^.83  ft.  For  100  ft.  from 
P.C.C.  the  offset  is  3.49— .29=3.20  ft.  This  is  a 
very  convenient  method. 

66.  Many     engineers     prefer    the     co-ordinate 
method.     The  circular  curve  is  run  from  the  P.C. 
established  by  making  the  offset  from  the  initial 
tangent,  and  the  spiral  is  then  located  by  setting 
off  ordinates  from  the  simple  curve  between  P.C. 
and    P.C.C.    and    by    ordinates    from    the    initial 
tangent  back  to  the  P.  S.,  or  for  the  latter  portion 
by   laying   off   o — y   from  a   tangent   at    the    P.C. 
parallel  to  the  initial  tangent,   the  ordinates  being 
calculated   by    one    of  the  preceding   methods  ;  or 
offsets  from  the  terminal    spiral  tangent   may   be 
made  for  the  last  third  of  the  spiral  length.     This 
method  is  particularly  applicable  to  location  work 
and  to  short  spirals,  though  under  many  conditions 
it  may  readily  be  applied  to  setting  track  centers. 

LAYING  OUT  THE  SPIRAL  BY  TRANSIT 
AND   DEFLECTION  ANGLES. 

67.  The  spiral  may  be  run  in  with  the  transit 
by  turning  off  deflection  angles  and  making  meas- 
urements along  chords  in  much  the  same  manner 
as    circular    curves.     The    deflection    angles    are 
easily  calculated,  and  the  field    work   is  not  more 
difficult   than    for  circular   curves.     The   ordinary 
transit-man  will  find  no  difficulty  in  understanding 
the   work.     Since  it  is  not  necessary  to  keep  sue- 


TRANSIT  AT  P.S.  43 

ceeding  chords  the  same  length  as  the  first,  the 
stationing-  may  be  kept  up,  and  the  even  stations, 
-f-50's,  and  other  points  put  in  as  usual.  Herein 
is  an  advantage  over  methods  requiring  a  regular 
length  of  chord  to  be  used. 

68.  Transit  at  P.   S.— With  the  transit  at  the 
P.S.,  which  has  been  located  by  one  of  the  methods 
previously  described,  the  deflection  angle  9  (BAL, 
Fig.  1,  page  6)  will  locate  points  on  the  spiral.      & 
may  be  taken  from  the  tables,   or  it  may  be  calcu- 
lated from  equation  ( 9),  9  =  ^  J  =  %  a  L\    For  this 
calculation,  if  desired,  the  square  of  L  may  be  taken 
from  a  table  of  squares,  the  lower  decimals  dropped, 
and  the  multiplication  by  the  simple  factors  remain- 
ing may  be  made  easily  and  rapidly.     Thus,  when 
a  =  2,  to  determine  6  for  a  point  234  ft.   (2.34  sta- 
tions)   from    the    P.S,     find    the    square    of    234 
(54756),  change  the  decimal  point  so  that  it  will 
become   the  square  of  2.34  (5.48),  and  &  =  \  a  U 
=  JX  2X5.48=1°  49'.     If  the  result  is  wanted  in 
minutes,    since    |X60  =  10,    use    10    instead   of  -J-. 
The  slide  rule  may  be  used  with  advantage. 

For  a  tabulated  spiral,  the  spiral  deflection 
angles  may  be  taken  from  the  table.  Thus,  for 
0  =  li,  by  Table  V  0  for  110  ft.  is  15 '.  For  114  ft. 
interpolate  proportionally  between  the  tabulated 
value  of  9  for  110  and  that  for  120  ft.  giving  14£'. 

If  it  is  not  desired  that  the  even  stations  be  lo- 
cated, the  spiral  may  be  located  by  50-ft.  chords,  or 
chords  of  other  length,  directly  from  the  P.S.  and 
the  labor  of  calculation  will  be  reduced. 

69 .  Transit  on  spiral, — With  the  transit  on  an  in- 
termediate point  on  the  spiral,  the    tangent  to  the 


44  LOCATION  BY  TRANSIT   ANGLES 

spiral  at  this  point  may  be  obtained  by  turning-  off 
from  the  chord  to  theP.S.  as  a  back-sight  the  angle 
A—  0  (ARF,  Fig.  2,  page  10),  where  A  is  the  spiral 
intersection  angle  and  9  is  the  spiral  deflection 
angle  at  the  P.S.  for  the  given  transit  point  (R). 
Except  for  extreme  lengths  this  is  equal  to  20. 
Thus,  by  Table  I  (a  =  £)  for  a  transit  point  400 
ft.  from  the  P.S.,  the  required  angle  is  4°— 1°  20' 
-=2°40r  (or  2  (1°20'). 

70.  For  deflection   angles  from  an  intermediate 
transit    point   on   ordinary    circular   curves,    three 
methods  are  in  use  among  engineers  : 

(a)  The  measurement  and  record  of  the  angle 
between   the  tangent  to  the  curve  at  the  transit 
point  and  the  chord  to  the  point  to  be  located. 

(b)  The   use  of  the  angle   between   the    chord 
connecting  the  transit  point  to  the  P.C.,   and   the 
chord  to  the  point  to  be  located. 

(c)  The  use  of  the  angle  between  a  line  through 
the  transit  point  parallel  to  the  initial  tangent  and 
the  chord  to  the  point  to  be  located.     Three  cor- 
responding methods  may  be  used  with  the    spiral 
and  will  be  treated  separately. 

71.  Intermediate  deflection    angles,     (a)    From 
tangent. — By  equation  (10),  the  angle  between  the 
tangent   at  the    transit   point  and    any   chord    (as 
CBH,    Fig.  4,  page  24)    is   *  =  &   a  L'  (L—  Z') 
±  \a  (L—Lfy.     (See  also  pages  11  and  23.)    This 
method  then  involves  the  following  steps  :     With 
transit  at  B  (Fig.  4,  page  24)  set  vernier  at  A' — 0' , 
or  2  0'  (these  being  the  angles  for  transit  at  the 
P.S.  for  the  point  B),   and  back-sight  on  the  P.S. 
so  that  the  zero  reading  will  give  the  tangent  BH. 


ANGLES   FROM    CHORD  45 

To  locate  any  point  C  find  the  sum  of  (1)  one  half 
of  the  product  of  the  degree-of-curve  at  the  transit 
point  B  by  the  distance  in  stations  from  the  transit 
point  to  C  and  (2)  the  spiral  deflection  angle  #  for 
the  same  distance.  For  D,  find  the  difference  of 
these  quantities.  Thus,  for  a  =  1,  with  the  transit 
300  ft.  from  the  P.S.,  D'  the  degree-of-curve  at  B 
is  3°.  By  Table  IV,  0'  (Z  =  3)  is  1°  30',  and 
ABG  is  3°,  giving-  the  position  of  the  tang-ent  at 
B.  For  C  100  ft.  from  B,  add  ^  X  3  X  1  =  1°  30' 
and  -J-  X  1  X  I3  =  10 ',  giving-  1°  40'  for  CBH.  For 
D  100  ft.  from  B,  DBG=  1°  30'— 10'  =1"  20'. 

72.  (b)  From  chord  to  the  P.S.— This  is  the 
method  g-enerally  to  be  recommended.  By  equation 
(11)  the  angle  between  the  chord  from  transit 
point  to  P.S.  and  any  chord  (as  CBE,  Fig-.  4,  pag-e 
24)  is  9  -f  \  D' L,  0  being-  the  spiral  deflection 
angle  from  the  initial  tang-ent  for  the  point  to  be 
located,  D'  the  degree-of-curve  at  the  transit 
point,  and  L  the  distance  in  stations  from  P,S,  to 
the  point  to  be  located.  (See  also  pag-es  11  and  24). 
This  method  involves  the  following-  steps  :  With 
transit  at  B  and  vernier  reading-  zero,  back-sig-ht  on 
the  P.S.  To  locate  C  turn  off  an  angle  equal  to 
the  sum  of  (1)  the  spiral  deflection  angle  B  for  a 
distance  equal  to  the  distance  from  C  to  the  P.S. 
and  (2)  one  sixth  of  the  product  of  the  degree-of- 
curve  at  the  transit  point  and  L  for  the  point  C. 
Thus  for  0  =  1,  with  the  transit  300  ft.  from  the 
P.S.,  D'  at  B  is  3°.  For  C  100  ft.  from  B  and  400 
1  ft.  from  the  P.S.,  add  \  X  1  X  42  =  2°  40'  (which 
is  the  spiral  deflection  angle  0  for  400  ft.)  and 
X  4  =  2°(whichisiZ>'Z)g-iving-4°40'for  CBE. 


46  LOCATION   BY    TRANSIT   ANGLES 

For  D  100  ft.  from  B,  DBA  =  40'  +  1°  =  1°  40'. 
To  facilitate  the  calculation  the  transit  point 
may  be  chosen  at  a  point  where  the  spiral  has  an 
even  degree-of-curve,  as  in  the  above  example,  but 
this  is  not  essential.  It  may  be  seen  thatTV^' 
gives  the  minutes  per  foot  in  ^  Df  L 

73.  (c)  Angles  with  initial  tangent    -The  use  of 
angles  with  the  line  parallel  to  the  initial  tangent 
(BK,  Fig*  4)  is  the  same  as  (b)  except  that  0' ,  the 
spiral  deflection  angle  to  the  transit  point,  must  be 
added  to  all  angles.     Otherwise  the  method  is  the 
same  as  (b).     Use  equation  (12). 

74.  Transit  at  P.O. C.— With  the  transit  at  the  P 
C.C.,  the  tang-ent  to  the  curve  may  be  found  by 
turning-  off  from  the  chord  to  the  P.S.    an  ang-le 
/Ji —  0i,  these  being*    the   angles  for  the  full  spiral. 
Within  ordinary  limits  this  ang-le  equals  2  #1.     The 
main  circular  curve  may  be  run  as  usual. 

In  case  the  P.S.  can  not  be  seen  from  the  P.C.C., 
the  chord  to  the  P.S.  may  be  located  by  turning-  off 
from  the  chord  to  an  intermediate  point  on  the 
spiral  an  ang-le  Jlr-0i— #  (ACB,  Fig-.  4,  page  24) 
where  0  is  the  ang-le  between  the  chord  and  the 
tang-ent  at  P.C.C.  (BCF)  (See  pag-e  11). 

To  locate  the  chord  from  P.C.C.  to  P.C.  (not 
shown  in  any  diagram),  deflect  from  the  chord  to 
the  P.S.  the  angle  J^i— #1.  To  locate  chord  to  P. 
C.  from  a  chord  to  an  intermediate  point  on  spiral, 
deflect  from  chord  to  the  intermediate  point  the 
ang-le  ^^i— $.  With  the  data  already  at  hand,  it 
may  be  easier  to  calculate  this  ang-le  as  #  +  ^  D' L — 
J-Ji-f-01,  remembering-  that  0  and  L  refer  to  the  inter- 
mediate point  and  D' ,  A,  and  0\  to  the  P.C.C. 


P.C.C.    TOWARD    P.S.  47 

75.  For  the  circular  curve  some  engineers  prefer 
to  measure  the  deflection  angles  from  the  tangent 
at  the  P.C.C.,  and  others  prefer  to  measure  from 
the  chord  from  P.C.C.  to  P.C.  and  thus   maintain 
the  same  notes  as  though  the  spiral  had  not  been 
used.     By  the  use  of  the  angles  discussed  in  pre- 
ceding paragraphs,  either  method  may  be  used. 

76.  To  run  from  the  P.C.C.   toward  the    P.S.- 
Two  methods  may  be  used,  (a)  using  L  as  the  dis- 
tance from  P.C.C.  and  deflecting  from  the  tangent, 
and  (b)  using  L  as  measured  from  the  P.S.  and  de- 
flecting from  chord  to  P.S. 

77.  (a)  Angles  from  tangent. — Using  the  distance 
L  as   measured  from  the  P.C.C.,  deflect   from  the 
tangent  to  the  curve  an  angle  equal  to  the  differ- 
ence of  (1)  one  half  of  the  product  of  D\  (degree-of- 
curve  at  P.C.C.)  and  distance  L   to  point  (which  is 
the  same  as  the  deflection  angle  for  D\°  circular 
curve)  and  (2)  spiral  deflection  angle  0  for  distance 
L,  \\aL^.     This  is  the   same   as   method    (a)   of 
"  Transit  on  Spiral."     The  method  depends  upon 
the  principle  that  the  spiral  deflects  from  the  oscu- 
lating curve  at  the  P.C.C.  at  the  same  rate  that  it 
deflects  from  the  initial  tangent  at  P.S. 

As  an  example  take  400  ft.  of  spiral  connecting 
with  a  4°  curve  (a  —  I).  Measure  L  from  P.C.C. 
For  a  point  150  ft.  from  P.C.C.  (Z  =  1.S),  take  the 
difference  between  ^  X  4  X  1.5  =  3°  and  22^  '  (spiral 
deflection  angle  for  150  ft.),  which  is  2°  37j£'.  This 
angle  is  to  be  deflected  from  the  tangent  at  P.C.C. 
By  the  same  method  the  angle  to  locate  the  P.S.  is 
y2  x  4  x  4^8°  minus  2°  40',  or  5°  20',  the  result 
found  by  the  usual  method. 


48 


LOCATION  BY  TRANSIT  ANGLES 


78  (b).  Angles  from  chord  to  P.S.— Using-  the 
distance  L  as  measured  from  the  P.S.,  deflect  from 
the  chord  to  the  P.S.  an  angle  equal  to  the  sum  of 
(1)  the  spiral  deflection  angle  0  for  distance  L  from 
the  P.S.  (^aL2)  and  (2)  one  sixth  of  the  product  of 
Di  (degree-of-curve  at  the  P.C.C.)  and  Z,  (£Z>iZ,). 
This  is  the  same  as  (b)  of  "  Transit  on  Spiral." 

Using  the  example  cited  in  the  preceding  method, 
150  ft.  from  P.C.C.  will  be  250  ft.  from  P.S.,  and  L 
=  2,5.  tf  =  iXlX(2.5)2=l02i'.  £.AZ=lx4 
X4  =  2°  40 ' .  The  sum  of  these  is  3°  42-| ' ,  the  angle 
to  be  deflected  from  the  chord  to  P.S.  By  the  same 
method,  for  P.S.  L  is  0,  and  the  deflection  angle 
proves  to  be  0,  as  it  should  be. 

79.  Transit  notes.—  For  aspiral  with  a=2  connect- 
ing with  an  8°  curve,  L  —  4,  and  if  the  P.S.  has  been 
found  to  be  at  16+29,  the  notes  may  be  made  as  fol- 
lows, using  method  (b)  for  the  transit  on  the  spiral. 
At  Sta.19  the  deflection  angle  from  the  chord  to  the 
P.S  ,  as  a  back-sight,is  the  sum  of  those  given  in  third 
and  fourth  columns,  and  it  is  here  inclosed  in  brackets. 

FIELD   NOTES 


5TA. 

POINTS 

6 

ial 

+  29o 

PCC.  6" 

5  ''20' 

J-37'     ( 
[6°57j  \ 

5f/  Vernier  at  <£  ~  -f°J56; 
bach-  5  ;<jh  t  <7/7  /#,  and  0  ' 

20 

4°  15~' 

J^/'l 
[7°56'1    • 

reading  q/ses  fart  ye  rtf. 

'+50 

J^6' 

TFJiT 

f€°  ?0j 

IS           ® 

0  '=  f.  V£ 

2*27' 

\ 

5ct  Vern/er  art  O?back-5/gfit 
on  PS.  and  forn  off  angle 

-f-50 

/'J7' 

~\ 

in  bracfiets 

/& 

o^s' 

/7 

o'/o  • 

Curve  to  ftwhf 

/6  -r-290 

&s 

0*0' 

a=^,   t~  =  4,  A^/6". 

TO  REPLACE  THE  ENTIRE  CURVE  49 

APPLICATION  TO  EXISTING  CURVES. 

80.  When  a  road  has  been  constructed  without 
transition  curves,   the  ordinary  application  of  the 
preceding-  principles  will  require  a  new  line  to  be 
built  inside  the  old  curve,  and  the  cost  of  construc- 
tion may  be  considerable.     To  retain  as  far  as  pos- 
sible the  old  roadbed,  three  methods  are  applicable  : 

(0)  To  replace  the  old  curve  with  a  new  and 
sharper  curve  located  so  as  not  to  vary  far  from  the 
old  alignment. 

(b)  To  replace  a  part  of  the  existing-  curve  with 
a  curve  of   slig-htly   smaller   radius,   compounding 
with  the  old  curve. 

(c)  To  make  a  new  alignment  for  the  main  part 
of  the  curve  close  to  the  old  and  replace  a   part  of 
this  with  a  curve  of  smaller  radius. 

81.  To  replace  the  entire  curve.     First  method. 
In  Fig-.  6,    the  dotted  line  TNH  is  the  old  curve, 
T  being-  its  P.C.     It  is  desired  to  throw  the  line 
out  at  H,  the  middle  point  of  the  curve,   a  distance 
HK  =p,  and  replace  the  curve  by  a  sharper  curve 
whose  P.C.  will  be  at  D,  thus  permitting-  the  spiral 
AEL  to  be  inserted.     P  is  the  intersection  of  tan- 
gents, which  comes  outside  the  diagram.     Let  R\  be 
the   radius   of  the  old   curve  and    R  of   the  new. 
HP—  KP  =p,  or 

R\  exsec  >£  I—(R+o)  exsec  */2  1  —  o  =•=/. 
Hence 

RI-R- 


exsec  *    vers  * 
—  R)  vers  %  I—o 

cos  y2  1 

=  (Ri—R—o)  exsec  */2  I—o  ......  (25) 


APPLICATION  TO  EXISTING  CURVES 


Also  AT  =  AP— TP  =  t—(Ri—R—o]  tan  J-  / 

*=t  —  (o+f)  cot  ft  I (26) 

by  which  the  P.S.  (A)  may  be  located  ;  or  if  T  is 
not  known,  the  tangent  distance  AP  may  be  calcu- 
lated and  A  located. 


T       B 


M 


FIG.  6. 


82.  Values  of/  from  zero  to  %  o  may  be  used. 
If  the  new  curve  comes  inside  the  old  at  the  center, 
p  must  be  used  as  negative  and  its  sign    in    the 
formula  must  be   changed.     It  must  be   borne    in 
mind  that  the  o  used  in  the  above  formula  must  be 
the  o  of  the  new  curve.     As  this  will  not  be  known, 
first  use  the  value  of  o  for  the  old  curve  in  (22), 
select  a  radius  and  degree  of  new  curve  near  the  re- 
sulting value,  and  then  determine  j>  and  AT  with 
the  o  for  the  new  curve. 

83.  As  an  example  take  /=  60°,  D  =  6°,  a  =  2. 
Then  o  for  a  6°  curve  is  3.93.     Take   1.0  ft.   as  a 


TO  REPLACE  THE  ENTIRE  CURVE  51 

trial  value  of  p.  By  equation  (24)  the  radius  of 
the  new  curve  will  be  approximately  35.8  ft.  shorter 
than  the  old  and  by  consulting-  a  table  of  radii  of 
curves  it  will  be  seen  that  a  6°  14'  curve  may  be 

used.     —  =  3.117  ;  there  will  be  311.7  ft.  of  spiral 
a 

at  the  end.  The  a  for  a  6°  14'  will  be  4.4  ft.  and 
the  resulting-/  is  found  by  (25)  to  be  0.5  ft.  There 
will  be  9°  43'  in  each  of  the  spirals  and  40°  34'  in  the 
remaining-  circular  curve  The  P.S.  maybe  located 
by  measuring-  the  tang-ent-distance  T,  or  the  middle 
point  K  of  the  curve  may  be  located  by  means  of 
the  external  distance,  E. 

84.  Second  method. — The  method  just  described 
may  be  modified  to  use  measurements  along-  the  ex- 
ternal secant  as  follows,  using-  Fig-.  6  as  before: 
Intersect  tang-ents  at  P.  (Intersection  outside  of 
diagram. )  Measure  PK  along*  external  secant  to 
the  point  K  where  it  is  desired  to  have  middle  point 
of  new  curve  come.  By  equation  (20)  (pag-e  18) 
calculate  the  radius  and  the  degree-of-curve  which 
will  g-ive  PK  as  the  external-distance  ^of  a  spiraled 
curve.  It  will  be  necessary  to  use  the  value  of  o  for 
a  degree-of-curve  equal  to  that  of  the  original  curve; 
since  the  degree  of  the  new  curve  is  not  yet  known. 
Next  select  a  curve  whose  degree  will  g-ive  a  radius 
close  to  that  found  by  the  above  calculation.  For 
this  D,  compute  o,  and  also  PK.  As  the  real  o  was 
not  known  in  the  first  calculation  and  the  new  curve 
will  not  have  exactly  the  R  found,  the  point  K  as 
now  located  may  not  coincide  with  that  first  chosen. 
Having-  located  K  anew,  the  curve  may  be  run  in 


52  APPLICATION   TO    EXISTING   CURVES 

from  K  with  back-sight  on  P,  or  the  tangent-dis- 
tances may  be  measured  to  locate  P.S. 

Instead  of  using  equation  (20),  PK  may  be  found 
by  adding  o  sec  ^  /to  the  external-distance  for  /°  of 
circular  curve  without  spiral.  Likewise  in  finding 
the  desired  D,  subtract  o  sec  |/from  the  measured 
distance  PK,  and  use  the  remainder  as  the  external- 
distance  for  an  un«piraled  circular  curve.  By  this 
means  a  table  of  external-distances  for  a  1°  curve 
may  be  utilized  and  the  calculations  shortened. 

85.  This  method   is   applicable  on  short  curves 
and  where  the  ground  will  permit  of  easy  and  ac- 
curate measurement  of  the  external-distance. 

Take  the  same  example  as  before.  Consider  that 
the  measured  PK  is  146.7  ft.  Using  o  =  3.93,  o  sec 
>^/=4.5  ft.  The  circular  curve  whose  external- 
distance  is  146.7—4.5  =  142.2  ft.  lies  between  6° 
14'  and  6°  13'.  Choosing  a  6°  14'  curve  and  re- 
calculating, the  external-distance  for  a  simple  curve 
is  found  to  be  142.2  and  o  sec  >^  /  5.1,  making  PK 
147.3  ft.  After  K  is  located  the  curve  may  be  run 
in. 

86.  (b)  To  replace  a  part  of  the  curve. — In  Fig. 
7,  B  is  the  P.C.  of  the  old  curve  whose  degree  is  Z>o. 
It  is  desired  to  go  back  on  this  curve  a  distance  BD 
and  there   compound   with   a   curve   of   somewhat 
sharper  curvature,  D\,  which  if  run  to  a  point  E 
where  its  tangent  is  parallel  to  the  original  tangent 
shall  be  at  a  distance  EF=  o  from  it.     The  tangent 
and  D\  curve  may  then  be  connected  by  a  spiral 
having  this  o.     It  is  required  to  locate  D  and  the 
P.S.  and  P.C.C.  so  that  a  selected  curve,  D\,  will 
give  a  calculated  or  assumed  distance  EF  as  o. 


TO   REPLACE    A   PART   OF 


Let  /?o  be  the  radius  of  the  .Z?o  curve  ancT^i  that 
of  the  Z>i  curve,  and  /i  the  angle  to  be  replaced. 


vers  /i— 


Having-  /i,  back  up  on  the  curve  to  D,  run  the  D\ 
curve  to  G,  the  P.C.C.  of  spiral,  and  locate  the  spiral. 


-On  final  Cur  re 


FIG.  7. 
The  P.S.  may  be  located  from  B  by 

AB=/— (/?0—/?r--0)tan  /i 

=t—(R, -A)  sin  7i.     (28) 

87.     Thus,  consider  that  a  part  of  a  4°  curve  is 
to  be  replaced  with  4°  30 '  curve,  and  that  0  =  1. 


54  APPLICATION   TO    EXISTING   CURVES 

o  =6. 62.  By  equation  (27),  h  =  16°  35'.  Takeout 
BD  — 414.6  ft.  of  4°  curve  and  locate  D.  16°35'  of 
4°30'  curve  requires  368.5  ft.  The  half  length  of 
the  spiral  is  225  ft.  The  P.C.C.  is  then  found  by 
running-  from  D  368.5—  225  —  143.5  ft.  of  4°30' 
curve  to  the  P.C.C.,  G.  Likewise  by  (28)  AB 
==224.8  —  45.4  =  179.4  ft. 

88.  The  limiting  values  of  D\  will  be  on  the  one 
hand  f  Z?o  and  on  the  other  a  value  which  will  make 
BD  one  half  the  length  of  the  original  curve.     Or- 
dinarily, D\  should  not  be  one  fifth  more  than  ZV, 
better  less  than  one  tenth  more  on  sharp  curves. 

89.  It  may  be  convenient  to  calculate  a  standard 
set  of  values  for  the  curves  on  a  road.     The  follow- 
ing gives  a' few  such  values. 

D*    a      Di      jRo  —  Rio         Ii         AB        GD 

2°  |  2°15'  318.3  3.31  8°16'  179.2  142.4 

2°  |  2°30'  572.9  4.53  7°12'  178.1  38.0 

3°  1  3°30'  272.8  3.12  8°40'  133.8  72.6 

4°  1  5°  286.4  9.07  14°27'  178.1  39.0 

5°  1  6°  190.9  15.65  23°2l'  223.4  89.2 

5°  2  6°  190.9  3.91  11°38'  111.4  43.9 

90.  (c)  To  re-align  and  compound. — When  the 
middle  portion  of  the  curve  is  in  fair  alignment  and 
it  is  desired  not  to  disturb  it,  or  when  it  seems  best 
to  re-locate  the  central  part  of  the  curve,  a  method 
by  taking  up  points  on  the  old  track  and  not  run- 
ning the  principal  tangents  to  an  intersection,  may 
be  used.     See  Fig.  8      Select  M,  N,  and  O  on  the 
curve    on   the   portion   not   to   be    disturbed.     Set 
transit  at  M,  measure  the  distances  MN  and  NO, 
and  by  the  usual  methods  for  circular  curves  deter- 
mine the  degree  of  curve,  D§,   which  will  fit  thi 


TO    RE-ALIGN    AND   COMPOUND 


55 


middle  portion.  Or  select  points  that  will  locate 
the  curve  in  a  desirable  position,  and  determine  DQ. 
The  selection  of  points  in  this  way  will  probably 
not  give  a  curve  whose  tangent  coincides  with  the 
track  tangent.  When  this  curve  is  run  back  until 
its  tangent  is  parallel  to  A  H  at  B,  the  distance 
from  the  track  tangent  will  be  called  m.  From  M, 
intersect  with  tangent  at  H  and  measure  /.  Deter- 
mine m  by  running  out  MDB  and  measuring  the 


FIG.  8 

offset  or  by  calculation  from  ra=HG  sin  /  and 
HG  =  HM— GM,  remembering  that  GM  is  the 
tangent-distance  for  7°  of  7?o  curve.  Let  ED  be  the 
new  D\  curve  which  must  be  run  in  from  D  so  that 
EF  shall  be  the  o  for  the  D\  curve.  Call  the  radius 


56  APPLICATION  TO   EXISTING  CURVES 

of  the  Z>o  curve  RQ  and  that  of  the  D\  curve  R\. 
The  D\  curve  will  have  I\  of  central  angle.     Then 
EK  =  o—  m  =  (RQ—Ri)  vers  /i. 


(29) 


If  o  is  less  than  m,  then  /?i  must  be  greater  than 
/?o.  If  K  comes  outside  of  AH,  m  must  be  added 
to  o.  Care  must  be  taken  that  M  is  far  enough 
back  on  the  curve. 

91.  To  locate  the  curves,   run  /—  /i  of  D§  curve 
from  M  to  D.     Run  DE  to  locate  the  P.C.,  or  run 
such  part  of  this  D\  curve  as  will  give  the  P.C.C. 
for  the  spiral. 

TheP.S.  (A)  may  be  located  as  follows  : 

AH  =  /  +  BG—  BK—  HG  cos  /  ........  (30) 

BG  =  GM,  the  tang-ent-distance  for  7°  of  D§  curve, 
BK  =  (/?o  —  /?i)  sin  /i,  and  HG  cos  /is  also  m  cot  /. 
The  P.S.  may  also  be  located  by  offsetting-  o  from 
E  to  F  and  measuring*  /  to  A. 

92.  For  example,  if  Do  has  been  found  to  be  4° 
and  /at  H  20°  and  m  1.2  ft.,  select  Z>i=4°  30  '  and 
a=l.     Then  o  =6.  62.     From  equation   (29)    /i  = 
15°.     Then  calculating-   the  length   of   the   curves 
from  the  ang-les  /and  /i,  as  MB  is  500  ft.  and  DB 
375  ft,    MD  —  125ft.      DE  =  333.3  ft.  since  there 
is  to  be  15°  of  4°  30  '  curve.     The  P.C.C.  for  spiral 
is  333.3—225  =  108.3  ft.   from.  D   toward  E,   since 
the  half  leng-th  of  the  spiral  is  225  ft.     AH  is  224.8 
+253.6—41.1—3.3  =  434.0  ft.     The  spiral  may  be 
run  in  by  usual  methods. 

93.  The   limiting-  values  of   D\   are   similar   to 
those  given  in  the   preceding-   method.     Generally 


METHODS   OF  TRACK    MEN  57 

D\  may  be  from  one  tenth  to  one  fourth  more  than 
Do,  depending-  upon  the  amount  of  the  curve  and 
its  degree. 

94.  When  the  new  D\  curve  is  so  much  sharper 
that  it  is  desired  to  connect  it  with  the  old  by  a 
spiral,  the  following-  method  is  applicable.  Call  o\ 
the  offset  to  tangent,  and  oo  the  offset  between  the 
two  curves,  the  latter  to  be  found  as  for  compound 
curves.  Then  by  a  method  similar  to  the  fore- 
going, 

T        o\  —  00  —  in 
vers  7i  = 


-  75—  - 

o  —  -t\.\  —  O  o 

95.  Methods  of  track  men.  —  When  curves  are  left 
without  transition  curves,  many  track  men  "ease" 
the  curve  by  throwing  the  P.C.  inward  a  short  dis- 
tance and  gradually  approaching  the  tangent  a  few 
rail  lengths  away,  while  the  main  curve  is  reached 
finally  by  sharpening  the  curve  for  a  short  distance. 

96-  Another  simple  method  for  track  which  is 
aligned  to  a  circular  curve,  consists  in  utilizing  one 
of  the  properties  of  the  transition  spiral.  In  Fig. 
1,  page  6,  let  ABK  be  the  original  track  line,  B  be- 
ing the  P.C.  Select  a  length  of  spiral  and  calculate 
o,  or  select  o  and  calculate  the  length,  by  a  preced- 
ing method.  At  a  distance  from  B  equal  to  half 
the  length  of  spiral  (point  on  the  curve  opposite  L) 
throw  the  track  inward  to  L  a  distance  equal  to  o. 
At  B,  the  old  P.C.,  throw  the  track  to  E,  a  distance 
half  as  great.  Measure  back  from  B  half  the  length 
of  the  spiral  to  A  for  the  beginning  of  the  ease- 
ment. Between  A  and  L,  line  the  track  by  eye,  or 
calculate  offsets  from  Table  IX.  The  remainder  of 


50  APPLICATION  TO  EXISTING  CURVES 

the  main  curve  must  then  be  thrown  in  the  same 
distance  as  at  L. 

97.  On  long-  curves  the  latter  work  would  be  ob- 
jectionable. It  may  be  avoided  by  using-  a  spiral 
running-  up  to  a  curve  whose  degree-of-curve  is  one 
third  greater  than  that  of  the  main  curve  and  com- 
pounding- directly  with  the  main  curve.  To  do  this, 
first  select  leng-th  of  spiral  for  a  curve  one  third 
sharper  than  the  circular  curve  which  call  L.  See 
Fig-.  9.  Call  the  circular  curve  Do  and  the  curva- 
ture of  the  end  of  the  spiral  D\.  Measure  back 
from  the  old  P.C.  on  tang-ent  a  distance  Y>  L,  which 


FIG.  9 

will  locate  the  P.S.  Measure  forward  on  the  curve 
from  the  P.C.  a  distance  \  L  to  locate  the  middle  of 
the  spiral,  and  offset  from  prolongation  of  tang-ent  a 
distance  equal  to  \  o,  or  \  o  from  the  circular  curve. 
Measure  also  along-  the  curve  from  the  P.C.  a  distance 


COMPOUND  CURVES  59 

f  L  to  the  P.C  C.  where  the  track  will  not  be  changed. 
The  spiral  will  pass  the  old  P.C.  at  /T  o  from  it,  and 
at  a  point  ^  L  from  the  P.C.C.  will  be  y&  o  distant 
from  the  circular  curve.  The  spiral  is  one  and  one 
half  times  as  long-  as  the  circular  curve  replaced. 

The  o  used  must  be  that  for  the  full  spiral  and  for 
the  sharper  curve,  f  /?o,  and  the  true  position  of  the 
circular  curve  should  be  known.  As  the  last  fourth  of 
this  spiral  is  sharper  than  the  main  curve,  the  eleva- 
tion of  the  outer  rail  up  to  the  P.C.C.  must  be  great- 
er than  that  on  the  main  curve,  gradually  reducing 
beyond  to  the  regular  amount. 

98.  Thus,  for  a  3Q  curve,  using  #—  1,  the  de- 
gree at  the  end  of  spiral  will  be  3  X  f  =  4,  and  the 
length  of  spiral  required  is  400  ft  ,  0  =  4.65.     The 
P.S.  will  be  133.3  ft.  back  of  the  P.C.,  the  middle 
of  spiral  66.7  ft.  ahead  of  P.C.,  and  the  P.C.C.  266.7 
ft.   ahead  of  P.C.     At  the  P.C.  the  track  must  be 
thrown  in  0.69  ft.,  at  the  middle  point  2.32  ft.  from 
tangent   (1.16    ft.    from   curve),   and  at  the   third 
quarter  point  .58  ft.  from  curve,  while  at  the  P.C.C. 
there  will  be  no  change.     Between  these  points  the 
track  may  be  aligned  by  eye,  or  ordinates  may  be 
calculated  by  Table  XII. 

However,  while  such  methods  are  easements,  they 
are  at  best  makeshifts  and  should  give  place  to  bet- 
ter methods. 

COMPOUND  CURVES. 

99.  The  spiral  may  be  used  to  connect  curves  of 
different   radii,  choosing  that  part    of    the    spiral 
having  curvature  intermediate  between  the  degrees 
of  the  two  curves  ;  thus,  connect  a   3°   and  an  8° 


6o 


COMPOUND  CURVES 


curve  by  omitting1  the  spiral  up  to  D  =  3°  and  con- 
tinuing- until  D  =  8°.  In  Fig-.  10,  DKM  is  a  Dl 
curve,  and  LNP  a  D^  curve,  the  two  curves  having 
parallel  tang-ents  at  M  and  N.  Di  is  greater  than 
D\.  Call  the  distance  MN  o.  It  is  desired  to  con- 
nect the  two  curves  by  a  spiral  shown  by  the  full 
line  KP.  The  degree  of  curve  of  the  spiral  at  K 
must  be  j9i,  and  at  P,  D^.  Consider  the  spiral  to 
be  run  backward  from  K  to  a  tangent  at  A.  Then 
the  spiral  from  K  to  P  is  the  portion  of  the  reg-ular 
spiral  from  where  its  degree  is  D\  to  the  point 
where  it  is  Z>2.  Since  the  spiral  diverges  from  the 
osculating-  circle  at  the  same  rate  as  from  the 
tang-ent  at  the  P.S.,  PN  =MK  and  the  spiral 
bisects  MN.  MN,  or  o,  is  the  offset  for  a  spiral  for 

H  B  A 


FIG.  10. 

a  curve  whose  degree  is  D% — D\.  Hence,  find  o  for 
a  Di — D\  curve,  and  make  the  offset  at  MN.  Meas- 
ure MK  and  NP  each  equal  to  >£  - 


-,    thus  lo- 


TO  INSERT  IN  OLD  TRACK  6l 

eating-  the  P.C  C.  of  each  curve  K  and  P.  Run  in 
the  spiral  from  K  or  P  by  the  method  for  point  on 
spiral  heretofore  described,  AK  being-  omitted. 
The  angle  between  tang-ents  at  K  and  P  is  A  for  a 
/>2  spiral  minus  J  for  D\  spiral,  and  may  also  be  ex- 
pressed as  Y*  (Z?i+  A)  times  KP  in  stations.  Thus, 
with#=2,  to  connect  a  3°  and  and  8°  curve,  o 
=  2.27,  the  value  for  a  5°  spiral.  The  portion  of 
the  spiral  used  will  be  250  ft.  long-.  Kis  125  ft. 
from  M,  and  N  is  125  ft.  from  P.  If  greater  ac 
curacy  is  required,  the  oc  COR.  for  this  leng-th  should 
be  subtracted  from  125.  The  angle  between  tan- 
gents at  K  and  P  is  ^  X2.50  (3°+8°)  =  13° 45' . 

The  spiral  may  also  be  used  to  connect  two  curves 
having  a  given  offset  between  them. 

100.  To  insert  in  old  track. — It  may  be  desired 
to  insert  a  spiral  between  the  two  curves  of  an 
existing  compound  curve  by  first  replacing  a  part 
of  the  sharper  curve  with  a  curve  of  slightly 
smaller  radius. 

In  Fig.  11,  let  AB  be  a  D\  curve  and  BG  a  D* 
curve,  B  being  the  P.C.C.  and  the  D%  curve  having 
the  smaller  radius.  Ci  is  the  center  of  the  D\ 
curve,  not  on  the  cut.  It  is  desired  to  go  back  on 
the  Z>3  curve  to  a  point  D  and  there  compound 
with  a  Z>2  curve  which  shall  be  run  to  a  point  E 
where  its  tangent  shall  have  the  same  direction  as 
the  tangent  to  the  D\  curve  produced  backward  to 
P  has  at  F.  The  radial  distance  EF  corresponds 
to  the  offset  of  the  usual  spiral  and  will  be  called 
o.  It  is  desired  to  locate  D  and  F  so  that  a  se- 
lected curve,  D<L,  will  give  a  calculated  or  assumed 
distance  EF  as  o. 


62  COMPOUND  CURVES 

The  distance  EF  is  made  up  of  FK  and  KE,  the 
first  being  the  divergence  of  the  D\  curve  from  the 
Z>3  curve  in  the  distance  BF  and  the  second  the  di- 
vergence of  the  Di  curve  from  the  D%  curve  in  the 
distance  DE.  Call  the  distance  BFZi,  and  DE,  £2. 
For  the  small  angles  used  these  divergences  may  be 
calculated  accurately  enough  by  the  approximate 
formula  for  tangent  offset,  y==.87  DU,  and  we 
shall  have 

EF=.87  (.A— Z>8)  £22+-87  (A— Z>i)  L?=o,  or 
(A— A)  Lf+  (Z>8— .A)  L?=  l.lSo . . .  (32) 


FIG.  11 

Since  the  amount  of  D\  curve    in  BF  plus    the 
amount  of  Di  curve  in  DE  (total  angle)  must  be 
equal  to  the  amount  of  D%  curve  taken  out,  we  have 
£><tL*+DiLi=Dz  (Z2+  Zi)  or 

(£>!—£>*')  Lz  =  (D*—  /?i)  Li (33) 

Combining  (32)  and  (33)  and  solving, 


TO  INSERT  IN  OLD  TRACK  63 


-,, 

(3S) 


101.  Having  L1  and  Z2,  the  points  D,  E  and  F 
may  be  located,  and  the  D^  curve  may  be  run  in 
from  D  as  far  as  necessary.  The  problem  is  then 
identical  with  that  of  putting-  a  spiral  between  two 
curves  having-  an  offset  o  (EF)  between  their  parallel 
tangents. 

By  the  principles  governing  the  placing  of  a 
spiral  between  two  curves,  it  is  seen  that  the  length 
of  the  connecting  spiral  L'  is  that  of  a  spiral  for  a 
curve  of  degree  equal  to  the  difference  of  degree  of 
the  two  connected  ;  that  is 


The  offset  is  equal  to  that  for  a  (Z>2  —  D^)  degree 
curve  from  a  tangent  or 

0  =  .0727  (Z>2  —A)  L'*  =  .0727  aLf*  ........  (36) 

Half  of  this  spiral  will  lie  on  one  side  of  the  offset 
and  half  on  the  other,  hence  in  Fig.  12  \L  to  the 
right  of  F  will  give  the  beginning  of  the  spiral,  H, 
and  ^  L  to  the  left  of  E  will  give  the  end  of  spiral,  I. 

1  02.     The  method  of  field  work  will  then  be  as 
follows:     Measure  from  B,  the  P.C.C.,    (Fig.    12) 
back  on  the  D±  curve  a  distance  BH  =  -JZ/  —  L\  to 
locate  the  point  of  spiral  H.     Measure  from  B  on 
the  DI  curve  the  distance  BD  =Ll  +  L^  to  D,  the 
new  P.C.C.,  run  in  the  £>2  curve  to  I,  DI  being  L^ 
+  Li  —  \L'  '  .     The  spiral  is  then  to  be  run  in  from 
H  to  I.     The  dotted  line  in  Fig.  12  shows  the  spiral. 

The  field  work  for  the  spiral  is  simple.  The 
spiral  may  be  run  in  by  offsetting  from  the  D^  curve 


64  COMPOUND  CURVES 

HF  (Fig-.  12),  knowing-  that  the  offset  from  the 
curve  to  the  spiral  is  the  same  as  that  of  a  spiral 
from  the  tang-ent  using-  the  distance  from  H  as  the 
distance  on  the  spiral.  Likewise  the  remainder  of 
the  spiral  may  be  offsetted  from  the  D2  curve  IE 
using-  distances  from  I  in  the  calculations. 

If  the  field  work  on  the  spiral  is  to  be  done  by 
deflection  ang-les,  the  spiral  may  be  run  in  from  H 
by  using-  as  deflection  ang-les  the  sum  of  the  deflec- 
tion ang-le  for  the  circular  curve  HF  and  the  spiral 
deflection  ang-le  from  a  tang-ent  for  the  same  dis- 


FIG.  12 

tance  ;  or  the  transition  spiral  may  be  run  backward 
from  I  in  a  similar  manner.  In  either  case  the  work 
will  be  no  more  difficult  than  for  spirals  for  simple 
curves. 


TO  INSERT  IN  OLD  TRACK  6$ 

1  03-  As  an  example  let  us  consider  that  a  2°  and 
an  8°  curve  are  compounded  at  B.  Consider  that  the 
degree  of  the  new  curve  to  be  run  in  is  8°  30',  and 
that  the  value  of  a  to  be  used  is  2.  Then  D\  =  2, 
J}B  =  8,  A  =  8  J.  For  a  spiral  from  2°  to  8°  30  '  ,  the 
value  of  the  offset  o  (EF)  is  the  same  as  the  o  for  a 
6°30'  curve  from  a  tang-ent.  Hence  0  =  4.99.  By 
formula  (34),  Z^  =.271,  and  by  formula  (35),  Lv=~ 
3.255.  Hence  the  point  D  will  be  back  on  the  Dl 
curve  325.5  +  27.1  or  352.6  ft.  from  B.  The  length 
of  the  spiral  to  be  used  will  be 


Of  this  162  5  ft.  will  be  to  the  left  of  E  and  162.5  ft. 
will  be  to  the  right  of  F.  Hence  H  and  I,  the  ends 
of  the  spiral,  may  readily  be  located  and  the  spiral 
may  be  run  in. 

1  04.  By  this  method  the  value  of  a  may  be 
chosen  beforehand,  the  value  of  o  may  be  easily  cal- 
culated, and  the  preliminary  field  work  is  small.  It 
may  be  stated  that  the  limiting-  values  of  D$  will  be, 
on  the  one  hand,  a  value  so  near  D$  that  the  result- 
ing" Li  will  carry  the  new  point  of  compound  curve 
back  to  the  end  of  the  old  curve,  and  on  the  other 
hand  such  that  the  leng'th  of  the  Di  curve  shall  be 
at  least  equal  to  half  the  length  of  the  transition 
spiral,  a  value  which  may  be  shown  to  be  Di  =  # 

(4J98  —  A). 

For  larg-e  angles  the  above  method  is  subject  to 
slig-ht  error. 


66  MISCELLANEOUS   PROBLEMS 

MISCELLANEOUS  PROBLEMS 

105.  To  change  tangent  between  curves  of  op- 
posite direction. — Having-  given  two  curves  of  op- 
posite direction  connected  b  j  a  short  tangent,  it  is  re- 
quired to  find  the  position  of  a  line  to  which  both 
curves  may  be  connected  by  spirals.     This  involves 
determining  the  angle  which  must  be  added  to  each 
curve  to  get  the  position  of  the  new  P.C.   of  each 
curve  for  spiraling-. 

106.  In  Fig-.  13  let  AB  be  the  original  tangent 
connecting  the  two  curves  and  /  its  length  in  feet. 


FIG.  13 

It  is  required  to  run  the  curve  KA.  to  E,  which  will 
be  the  P.C.  for  the  spiraled  curve,  and  MB  to  F  for 
its  spiraled  P.C.,  and  also  to  find  the  position  of  the 


CURVES   OF   OPPOSITE    DIRECTION  67 

line  CD,  which  will  be  the  common  tangent  for  the 
two  spirals.     Call  the  angle  APC  a.     It  is  the  same 
as  that  of  the  additional  amount  of  curve  AE  and 
BF.     Let  A  be  the  radius  of  the  curve  KE  and  Ri 
that  of  MF,  and  o\  and  0-2  be  the  respective  spiral 
offsets  NE  and  OP  for  the  spirals  chosen  for  the  two 
curves.     AC  +  BD  =  NE  +  EG  +  OF  +  FH. 
Then,  since  AC  +  BD  =  /  sin  «  and  EG  =  Ri  vers  «, 
etc., 

/sin  a  =  0i  +  02+Cffi  +  /?2)  versa  ...........  (37) 

107.  Since  a  will  be  small,  we  may  substitute, 
using-  «  in  degrees,  sin  «  =  .  01745  a  and  vers  «  — 
.000  152  «2,  which  are  close  approximations  below 
8°.  Transforming, 

01  +  02        .000152  (A+  A)    2 
" 


".  01745  /"  .017457 

This  quadratic  may  be  solved,  but  usually  the  fol- 
lowing approximate  root  gives  sufficiently  close  re- 
sults : 

= 


-  (.01745  /)• 

Having  a  the  lengths  AE  and  BF  may  be  found, 
the  position  of  the  P.C.C.  of  each  curve  found,  and 
the  new  tangent  located  by  offsetting  EN  and  FO, 
or  by  offsetting  AC  (equal  to  01  -f/?i  vers  «)  and  BD 
For  very  short  tangents,  spirals  must  be  chosen 
short  enough  not  to  overlap  on  the  tangent. 

108.   As  an  example  take  a  3°  curve  and  a  4°  curve 
connected  by  600  ft.  of  tangent.     Use  a  =  1,     Then 
01  -  1.96  and  01  =4.65.     By  equation  (38) 
«  =  .63  +.02—  .65°  -  0°39'. 

This  result  checks  equation  (37)  very  closely.  0°39' 
gives  21.7  ft.  of  3°  curve  (AE)  and  16.2  ft.  of  4° 


68  MISCELLANEOUS   PROBLEMS 

curve  (BF).  There  will  be  300  ft.  of  spiral  for  the 
3°  curve  and  400  ft.  of  spiral  for  the  4°  curve.  The 
P.S.  and  P.C.C.  of  each  spiral  will  be  half  of  the 
spiral  leng-th  from  the  points  E  and  F.  The 
P.C.C.  of  one  will  be  (150  —  21.7=128.3)  ft.  back 
of  A,  and  of  the  other  (200—16.2  =  183.8)  ft.  back 
of  B.  AC  will  be  (1.96  +  .11  =-2.07)  ft.  and  BD 
(4.65  -f- .09  =-4.74)  ft.  The  distance  from  C  to  P.S. 
will  be  150 -f  21.7,  and  from  D  to  the  other  P.S. 
200  +  16.2,  neglecting*  the  /correction.  The  spirals 
may  then  be  run  in  as  usual, 

109.  This  solution  may  also  be  applied  to  the 
case  where  a  tang-ent  thrown  off  from  the  curve  KA 
does  not  strike  the  curve  MB  but  is  parallel  to  this 
curve  at    a   point   opposite   B  distant  m  from   it. 
Since  cos  «  is  nearly  1,   equations   (37)   and   (38) 
may  be  modified  by  subtracting-  m  from  (01  +  02) 
wherever  it  occurs.     This  modification  is  of   con- 
venience in  revising-  old  lines.     The  engineer  should 
make  his  own  diagram. 

110.  To  change  tangent  between  curves  of  same 
direction. — Having-   given  two  curves  of  same   di- 
rection connected  by  a  tang-ent  it  is  desired  to  find 
the  position  of  a  line  to  which  the  two  curves  may 
be  connected  by  spirals.     As  in  the  preceding-  prob- 
lem this  involves   determining-  the  chang-e  in  the 
angle  of  the  two  curves  and  the  position  of  the  P. 
C.  of  each  curve  for  spiraling-. 

111.  In  Fig-.  14  let  AB  be  the  original  tang-ent 
connecting-  the  two  curves  and  /  its  leng-th  in  feet. 
It  is  required  to  back  up  on  the  curve  AK  to  E  for 
the  P.C.   for  spiraled  curve  and  to  run  the  curve 
MB  to  F  for  its  spiraled  P.C.,  and  to  find  the  po- 


CURVES   OF   SAME    DIRECTION 


sition  of  the  line  CO  which   will  be  the  common 

tang-ent  for  the  two  spirals.     Call  the  angle  BAD'  «. 

Aten/  Tangent  *v 

C  \  O 


N 


O 


FIG.  14. 

It  is  the  same  as  that  in  AE  and  BF.  Let  R\  be 
the  radius  of  the  curve  KE  and  /?2  that  of  MF,  and 
o\  and  02  be  the  respective  spiral  offsets  NE  and 
OF  for  the  spirals  chosen  for  the  two  curves 

BD— AC  =.  OF  +  FH— NE— EG. 
Then,  since  BD— AC  or  BD'  equals  /  sin  «  and 
EG  equals  R\  vers  «,  etc., 

/  sin  a  =  02  —  01  —  (Ri—Rz)  vers  a (39) 

1  12.  Since  «  will  be  even  smaller  than  in  the  preced- 
ing problem,we  may  substitute  sin  «  =  .01745  «  and  vers 
a  =  .000152  a2,  using-  « in  degrees.  Transforming-, 

oi  —  oi     __  .000152  (/?i— A)  aa 
"  . 01745 /  "  .01745  / 

As  in  the  preceding-  problem,  the  approximate 
solution  of  this  quadratic  may  be  used. 


70  MISCELLANEOUS    PROBLEMS 

= 


-017457-  (.01745  0* 

The  last  term  here  is  very  small. 

Having*  «  the  lengths  AE  and  BF  may  be  found, 
the  position  of  the  P.C.C.  of  each  curve  found,  and 
the  new  tangent  located  by  offsetting-  EN  and  FO, 
or  by  offsetting-  AC  (equal  to  o\  +  R\  vers  «)  and 
BD  (equal  to  02  +  R<L  vers  «).  The  length  of  spiral 
must  not  be  so  great  that  the  spirals  will  overlap 
on  short  tangents. 

113.  As  an  example  take  a  3°  curve  and  a  4° 
curve  connected  by  600  ft.  of  tangent.  Use  a  =  1. 
Then  o\  =  1.96  and  02  =  4.65.  By  equation  (40) 
«-  .257  —  .006  =  .251  =  0°  15J'. 

0°  15^'  gives  8.6  ft.  of  3°  curve  (AE)  and  6.4  ft. 
of  4°  curve  (BF).  There  will  be  300  ft.  of  spiral 
for  the  3°  curve  and  400  ft.  for  the  4°  curve.  The 
P.C.C.  of  spiral  will  then  be  (150  +  8.6  —  158.6) 
ft.  back  of  A  and  (200  —  6.4=193.6)  ft.  back  of  B. 
AC  will  be  1.98  ft.  and  BD  4.67  ft.  The  distance 
from  C  to  P.S.  will  be  150  —  8.6  and  from  D  to  the 
other  P.S.  200  +  6.4,  neglecting  the  /correction. 
The  spirals  may  then  be  run  in. 

114.  This  solution  may  also  be  applied  to  the 
case  where  a  tangent  thrown  off  from  the  curve  K  A 
misses  the  curve  MB  by  a  distance  m  from  B  the 
point  of  parallelism.  In  this  case  equations  (39) 
and  (40)  may  be  modified  by  subtracting  m  from 
(02  —  01)  wherever  it  occurs.  If  the  second  curve 
is  one  of  larger  radius,  it  will  be  necessary  to  con- 
struct a  new  diagram  and  determine  the  signs  of 
the  terms. 


UNIFORM  CHORD  LENGTH  METHOD  7l 

UNIFORM  CHORD  LENGTH  METHOD. 

115.  The  treatment  of  the  spiral  heretofore 
given  is  based  upon  principles  which  permit  the 
use  of  any  chord  length,  either  uniform  or  variable, 
throughout  the  length  of  the  spiral.  Regular  chord 
lengths,  like  20  or  25  feet,  may  be  used,  if  de- 
sired, and  the  excess  if  any  used  as  a  fractional 
chord  at  the  beginning-  or  the  end  of  the  spiral. 
If  it  is  desired  to  use  chords  of  common  length, 
another  method,  known  as  uniform  chord  length 
method,  may  be  derived  by  modifying1  the  preceding 
formulas.  A  further  modification  of  this  method 
may  be  made  to  allow  the  use  of  fractional  chord 
lengths  at  the  beginning  or  the  end  of  the  spiral,  so 
that  it  will  not  be  necessary  to  make  the  uniform 
chord  length  an  aliquot  part  of  the  length  of  the 
spiral.  Thus,  if  the  spiral  is  to  be  203.2  ft.  long, 
ten  20-ft.  chords,  or  eight  25-ft.  chords,  or  thirteen 
15-ft.  chords,  etc.,  may  be  used — the  first  or  last 
chord  or  both  being  fractional. 

1  1 6.  The  notation  used  will  be  the  same  as 
heretofore  except  as  noted,  and  the  equations  will 
be  numbered  the  same,  using  the  prime  mark  to 
distinguish  them.  Let  c  be  the  chord  length  used. 
This  will  be  expressed  in  hundreds  of  feet,  that  is 
in  the  number  of  100-ft.  stations.  For  a  chord 
length  of  20  ft.,  c=.2]  for  one  of  15  ft.,  £  =  .15, 
etc.  Let  n  (an  integer)  represent  the  number  of 
full  chords  from  the  P.S.  to  a  desired  point.  In 
Fig.  15,  A  is  the  P.S.  and  its  n  is  0.  The  n  of  B 
is  1,  of  C  2,  etc.  Let  Q\  =  spiral  deflection  angle 
at  P.S.  from  initial  tangent  for  a  single  full  chord 


72  UNIFORM   CHORD   LENGTH    METHOD 

length  (BAE)  (called  unit  spiral  deflection 
angle),  and  On  for  n  chord  lengths.  For  D  n  -~3 
and  On  =:  DAE.  Similarly,  Jn,  Dn,  and  Lw  are  for  a 
point  «  chord  lengths  from  the  P.S.  For  the  in- 


\\\ 


K 


FIG.  15. 

strument  at  other  points  than  P.S.,  let  n'  be  the 
number  of  chord  lengths  from  the  P.S.  to  the  chord 
point  at  which  the  instrument  is  located,  reserving 
n  still  as  the  number  of  chord  lengths  from  the  P.S. 
to  the  point  to  be  located,  and  let  $n  represent  the 
deflection  angle  from  tangent  at  the  instrument  point 
to  this  desired  point.  Thus,  if  the  instrument  is 
at  C  two  chord  lengths  from  the  P.S.,  n'  =  2,  and 
to  locate  D  three  chord  lengths  from  the  P.S.,  n  =  2> 
and  $n  will  represent  the  deflection  angle  DCK  to  lo- 
cate the  third  chord  point  D. 

117.  The  length  L=nc.  By  substitution  in  equa- 
tions (1),  (2),  and  (9)  of  the  spiral,  we  have  for  any 
point  on  the  spiral  distant  L=nc  from  the  P.S. 


UNIFORM  CHORD  LENGTH  METHOD  73 


\an*<*  ........................  (2f) 

For  end  of   first  full  chord  and  for  end  of  n  full 
chords,  respectively 


(9') 


(10') 


In  formula  (10'),  the  arithmetical  difference  of 
the  numbers  of  the  chord  points  is  taken,  rather 
than  the  algebraic  difference.  If  the  latter  is  used, 
the  signs  of  operation  should  all  be  plus. 

118.  The  first  step  is  to  calculate  the  value  of 
the  unit  spiral  deflection  angle  0i  by  means  of  eqna- 
tion  (9'),  using-  a  and  c  or  other  terms.  For  a 
chord  length  of  20  ft.,  and  a  value  of  a  =  2,  c  =  .2 
and  0j.  =  ^  X  2  X  (2)2  =--0°0.8'.  If  a  spiral  250  ft. 
long  is  to  connect  with  a  4°  curve,  using  25-ft. 
chords,  Z?n  =  4  and  Ln  =  2.5,  0{  =  tf  x  4  X  TV  X  275 
^0°lr.  If  4i  =  9°,  Z—  3  and  r—  .2  (20  ft.),  ^  = 

t    (|)3X90^0>8^ 

1  1  9.  The  value  of  0i  gives  a  basis  for  computing 
the  deflection  angle  for  other  points  ;  thus  for  a 
point  5  chord  lengths  from  the  P.S.,  n  =  $  and  the 
deflection  angle  by  equation  (9')  is  25  times  the 
value  of  0i-  For  the  instrument  at  the  fifth  chord 
point  (n'=  5),  the  deflection  angle  from  the  tangent 


74      \  UNIFORM  CHORD  LENGTH  METHOD 

at  the  instrument  point  to  a  point  8  chord  lengths 
from  the  P.S.  («  =  8)  is  by  equation  (10')  :  $n  = 
[3  X  5  (8  -  5)  ±  (8— 5)8]  0i  =  54  0i,  To  locate  from 
the  same  instrument  point  a  point  3  chord  lengths 
from  the  P.S.  (2  from  the  instrument  point),  the 
deflection  angle  is  26  0i. 

120.  Table  of  unit  spiral  deflection  angles.— A 
table  giving  Q\  for  various  chord  lengths  for  many  of 
the  values  of  a  used  in  field  work  may  be  of  service. 
Table  XIII  gives  spiral  deflection  angles  for  first 
chord  length  (unit  spiral  deflection  angles).    The 
angle  is  given  in  minutes.     It  is  well  in  the  calcu- 
lations to   express  decimals  as  common  fractions  ; 
thus,    for  20- ft.    chords  with  a=-\\  use  61=.$%'; 
for  16-f t.  chords  with  a  =  If  use  Ol  = .  42f ' . 

121.  Table  of  coefficients  for  deflection  angles 
The   values   obtained   from  (9')  and  (10')  may  be 
considered     as    coefficients    of    #1,    and    a  general 
table    prepared.       Table    XIV    is    a    table  of   co- 
efficients for  a  spiral   up  to   15   chord  lengths  for 
use  with  the  instrument  at  any  chord  point. 

122.  For  the  instrument  at  the   P.S.,  multiply 
the  coefficient  in  the  column  headed  0  opposite  the 
chord  point  to  be  located  by  the  value  of  the  spiral 
deflection  angle  for  a  single  chord   length    (unit 
spiral  deflection  angle  0i). 

123.  To    find    the    deflection    angle    from     the 
tangent  at  any  chord  point,  enter  the  column  whose 
heading  gives  the  number  of  the  chord  point  at 
which  the   instrument  is  placed  and  take  the  co- 
efficient opposite  the  number  of  the  chord  point  to 
be    located;  then    multiply    the   spiral    deflection 
angle  for  a  single  chord  length   (0i)  by  this  coeffi- 


FRACTIONAL  CHORD  L 

cient.  Thus,  as  in  example  cited  above™ Tor  the  in- 
strument at  5,  the  deflection  angle  from  the  tangent 
at  this  point  to  locate  a  point  8  chord  lengths  from 
the  P.S.  is  found  to  be  54  #1,  and  to  locate  a  point 
3  chord  lengths  from  the  P.S.  is  26  Blt  This  table 
may  easily  be  extended.  The  variation  in  the 
tabular  differences  in  horizontal,  vertical,  and 
diagonal  directions  is  readily  discerned,  and  if  pre- 
ferred the  method  of  differences  may  be  used  for 
calculating  deflection  angles  for  a  particular  case 
in  place  of  a  multiplication  of  these  coefficients. 

124.  To  end  the  spiral  with  a  fractional  chord. — 

If  the  number  of  chord  lengths  is  not  integral,  the 
first  and  succeeding*  chords  may  be  made  of  uniform 
length  until  the  last  one  is  reached  and  the  full  de- 
flection angle  may  be  turned  off  for  the  P.C.C. 
Thus,  for  218.4  ft.  of  spiral,  ten  20-ft.  chords  may 
be  used  and  the  full  deflection  angle  turned  off  for 
the  remaining  18.4  ft. 

125.  To  begin  the  spiral  with  a  fractional  chord 
length. — In  case  it  is  desired  to  begin  the  spiral 
with  a  fractional  chord  length,  the  following  modi- 
fication may  be  made.     Let  m  be  the  ratio  of  this 
fractional  chord  length  to  a  full  chord  length,  and 
0m  be  the  spiral  deflection  angle  from  the  initial 
tangent  for  this    fractional    chord  length,    which 
from  the  general  formula  for  0  may  be  seen  to  be 
m*  0i.     Let  0n  +  m  be  the  spiral  deflection  angle  from 
initial  tangent  to  locate    a    point   (n  +  m)   chord 
lengths  away  (n  an  integer  and  m  fractional),  and 
$n-f  m  the  deflection  angle  from  tangent  at  instru- 
ment point   (n'  +  m)   chord  lengths  from  P.S.  to 


76  UNIFORM  CHORD  LENGTH  METHOD 

locate  a  point  (n  +  m)  chord  lengths  from  P.S. 
Substituting-  in  formula  (9)  page  9, 

i=  (n*+2nm+m*)9i 

.  .  .  .  (  9  '  ) 

126.  In  the  last  member  of  equation  (9"),  the 
first  term  is  the  spiral  deflection  angle  for  n  full 
chords,  the  second  term  is  n  times  a  constant,  and 
the  third  term  is  the  spiral  deflection  angle  for  the 
fractional  chord.  The  calculations  may  be  simpli- 
fied by  the  method  of  differences. 

For  example,  for  a  chord  length  of  20  ft.  let  the 

beginning  chord  be  8.4  ft.     Then  m  =  -^.  =.42.    If 

1.2'  be  the  unit  spiral  deflection  angle  #1,  Om  =.21  '  . 
To  locate  a  point  88.4  ft.  from  P.S.,  the  spiral  de- 
flection angle  at  P.S.  will  be 

#»+m=  16xl.2-f4X2X.42X.2i  +  .2=-=23'.4. 

Jse  Table  XIV  in  calculating  On  . 

1  27.  For  the  instrument  at  a  chord  point 
(nr  +  m*)  chord  lengths  from  the  P.S,  the  deflec- 
tion angle  from  the  tangent  at  this  chord  point  to 
locate  a  chord  point  (n  +  m)  chord  lengths  from 
the  P.S.  is  found  from  equation  10  page  12  to  be 


=  *n  +  («—»')  (30*00  ............  (10*) 

In  the  last  member  of  equation  (10")  the  first 
term  is  the  deflection  angle  for  full  chords,  and  the 
second  term  is  (n  —  n'}  times  a  constant.  If  n'  is 
greater  than  ^,  the  second  term  is  still  added 
numerically  to  the  first.  Tabular  differences  may 
also  be  used. 


DEFLECTION    ANGLES  77 

128.  To  use  equation  (10")  first  calculate  #n, 
using-  Table  XIV  ;  then  add  the  last  term.  Thus, 
for  a  chord  length  of  20  ft.  and  a  beginning-  chord 
of  8.4  ft.  and  a  unit  spiral  deflection  angle  of  1.2', 
with  the  instrument  88.4  ft.  from  P.S.,  n'  +m  = 
4.42.  To  locate  a  chord  point  148.4  ft.  from  P.S. 
(n  —  7*),  the  deflection  angle  from  the  tangent  is 
found  to  be,  taking  the  coefficient  for  ^from  the 
column  headed  4. 

0*+,w  =  45X  1.2'  H-  3X3X.42X  .21'  =54.8'. 

To  locate  a  chord  point  48.8  ft.  from  P.S.,  using 
the  same  instrument  point,  (n  =  2,  n'  —  4,  m  —  .42), 
we  have  for  the  deflection  angle 

0»  +  m  =  20  X1.2'  +2  X  3  X.  42  X  .21'  =  24.5'. 

1  29.  To  illustrate  the  use  of  these  methods,  take 
the  following  examples.  It  is  desired  to  spiral  a 
6°  40'  curve  with  200  ft.  of  spiral,  using  20-ft. 
chords,  c  —  .20.  n  =  10.  Dn  —  6f  .  By  equation 
(9')  the  spiral  deflection  angle  0\  for  a  20-ft.  chord 

9Q2 

is  i  X  '-^-X6°  40'  =  H'  =  0i.     The  multiplication 

of  !£'  by  the  coefficients  in  Table  XIV  for  instru- 
ment at  0  and  for  instrument  at  10,  gives  the  de- 
sired deflection  angles. 

If  it  is  desired  to  connect  a  tangent  with  a  4° 
curve  so  that  the  offset  o  shall  be  5.0  ft.,  proceed  as 
follows.  By  equation  (14)  the  length  of  spiral  is 
415.2  ft.  Using  25-ft.  chords,  the  deflection  angle 


in  minutes  is  by  equation  (9  ')  —          *    =.602  —  #1. 

T".  J-O^w 

Table  XIV  will  give  the  coefficients  for  multiplica- 


78  UNIFORM    CHORD    LENGTH    METHOD 

tion,  and  the  fractional  chord  may  be  left  for  the 
last  measurement. 

130.  To  show  the  use  of  fractional  beginning- 
and  ending*  chords,  consider  that  a  spiral  138.4  ft. 
long*  is  to  connect  with  a  10°  curve  and  that  it  is 
desired  to  use  15-ft.  chords  but  that  the  first  chord 


shall  be  9.3  ft.     £  =  . 


9-3  tm       TD 

= -r-£-~=  .62.    By  equa- 


tion  (9'),  the  spiral  deflection  angle  for  a  15-ft. 
chord  #1  is  found  to  be  1| ',  and  for  the  point  9.3  ft. 
from  P.S.0m  is.622  X  1|=.62'.  The  table  below 
gives  values  for  field  work,  considering-  that  P.S.  is 
at  Sta.  322  +  13.7.  In  the  column  headed  "number 


No.  of 

INSTRUMENT  POINT. 

Survey 

rVinrrl 

CVntral 

Station. 

Point. 

Angle. 

P.S. 

0.62 

1.62 

4.62 

5.62 

322+13.7 

P.S. 

o 

o 

1.2' 

I°OQ4' 

+23- 

0.62 

1.8' 

0.6' 

o 

i°O4.i  ' 

+38. 

1.62 

12.8' 

4-3' 

4.6' 

O 

53-0' 

4-53- 

2.62 

33.5' 

11.2' 

12.$' 

38.6' 

+68. 

3.62 

i°03.9' 

21.3' 

23.7' 

20.9' 

+83- 

4.62 

i°44.i' 

34-7' 

o 

+98. 

S.62 

2°34.o' 

51.3' 

24.1 

o 

323+13. 

6.62 

3°33-6' 

I°II.2' 

+28. 

7.62 

i°34.3' 

+43- 

8.62 

2°00.7' 

+  52.1 

End. 

6°55-3' 

2°i8.4' 

of  chord  point"  the  integ-er  of  the  number  is  ;/  and 
the  fractional  part  is  m.  In  the  succeeding- 
column  heading's  for  the  instrument  points,  the  in- 
teger is  ri .  Thus,  to  determine  the  deflection 
angle  with  the  instrument  at  0  to  locate  323  X  43, 
by  equation  (9")  and  Table  XIV, 

0n+m  =  (64x  1^) +  8X2.01)  +.6-2°  00.7/. 


UNIFORM  CHORD  LENGTH  METHOD  79 

To   determine  the  deflection  with  instrument   at 

322  +  83,   (>'  =4),  to    locate  322  +  68  («  =  3),  by 
equation  (10")? 

0»+m  =  (llX  1|)  +  (1  X  3.02) -20.9'. 

131.  To    run    in  the  spiral  from    the     P.C.C., 
this  method  may  likewise  be  used  if  the  P.C.C.   is 
at  n  or  at  n  +  m  chord  lengths  from  the  P.S.     If  it 
is  not,  that  is,    if   both    the   first   and   last   chord 
lengths  are  to  be  fractional,  the  points  on  the  spiral 
as  far  as  the  next  instrument  point  may  be  set  by 
the  principle  that  the  deflection  angles  will  equal 
the  difference  between  the  deflection  angle  for  a 
circular    curve    and    the    deflection    angle    for    a 
spiral  from  initial   tangent,   both   for   a    distance 
equal  to  the  distance  to  the  desired  point.     After 
the  next  instrument  point  is  reached,  calculate  de- 
flection angles  as  though  working  from  the  P.S. 

132.  The  method  of  uniform    chord  lengths  is 
subject  to  the  same  correction  for  9  as  is  given  on 
page  10.     As  it  is  not  likely  that  this  method  will 
be  used  for  large  deflection  angles,  the  error  will 
usually  be  negligible.     For  $  the  correction  needed 
is  almost  exactly  that  for  the  0  which  enters  into 
it ;  thus  in  equation  (10')  make  a  correction  which 
would   be  necessary  for  a  0  equal   to  (n — #')80i. 
This  is  not  often  necessary. 

1 33.  It  will  be  seen  that  the  method   of  uniform 
chord  lengths  may  have  advantages  where  a  chord 
of  full  length  may  be  used  at  the  beginning  of  the 
spiral,    especially    where   the  rate  a  is  fractional, 
and  that  it  is  also   applicable  when  the  beginning 
chord  is  fractional.     It  is  more  especially  applicable 


80  STREET   RAILWAY  SPIRALS 

where    evenly   spaced   points   are   wanted.     Table 
XIV  is  a  convenient  table. 

STREET  RAILWAY  SPIRALS. 

1 34.  For  use  in  connection  with  curves  of  short 
radii,  as  street  railway  curves,  the  formulas  for  the 
transition  spiral  may  be  modified  with  advantage. 
The  variable  radius  R  of  the  spiral  may  replace  the 
degree  of  curve  D.     The  product  of  the  radius  at 
any  point  by  its  distance  from  the  P.S.   will    be 
shown  to  be  constant  for  a  given  spiral,   and  this 
product  may  be  used  as  the  characteristic  constant, 
taking-  the  place  of  a.     The  offset  o  may  be  used  as 
one  fourth  of  the  ordinate  y  of  the  terminal  point 
of  the  spiral  except  for  extreme  lengths.     Certain 
other  approximations  may  be  made  which  are  not 
always  allowable  with  curves  of  large  radius. 

135.  Theory. — The  general  notation  will  not  be 
changed.     Fig.  16  shows  the  co-ordinates  x  and  y, 


RCO 


FIG.  16. 


spiral  intersection  angle  4,  spiral  deflection  angle  0, 
and  spiral  tangent-distances  u  and  v  for  a  point  on 


FORMULAS  8l 

the  spiral,  and  also  o  and  t  for  the  full  spiral,  to- 
gether with  the  produced  circular  curve.  As  be- 
fore, R  =  radius  of  curvature  at  any  point  and 
5=  length  of  the  spiral  arc  in  feet  from  P.S.  to  any 
point  of  the  spiral.  From  equation  (1)  page  7, 

D       100 D       573000 


= 


L~       s  sR 


573000 

Hence        5  R  =  —      —  ,  a  constant. 
a 

Represent  this  constant  product  of  5  and  R  by  k. 
Then 

,          D      573000       A   „        k  /K1N 

k  =  sR  =  —       -  and  R  =  —  .  .  ..........  (51) 

a  s 

The  property  of  the  transition  spiral  that  R 
varies  inversely  as  the  distance  along  the  spiral  is 
satisfied  by  this  equation. 

1  36  Modifications  of  the  formulas  already  de- 
rived may  be  made  as  follows.  The  angle  sub- 
tended by  a  circular  arc  in  degrees  is  equal  to 

180     5        57.3^      0.  ., 

--  —  =  —  •=  —  .     Since  A  is  one  half  as  much  as 

7T          J\  J\ 

the  angle  of  the  same  length  of  circular  curve 
having  a  radius  equal  to  the  terminal  radius  of  the 
spiral, 

A  _  28.655  =  28.65s2  ,  g  . 

This  may  also  be  derived  directly  from  equation  (2) 
page  7  by  substitution  for  the  values  of  a  and  L* 
Also 


82  STREET   RAILWAY    SPIRALS 

For  large  angles,  if  the  precise  values  of  0  are  de- 
sired, the  corrections  given  in  the  table  on  page  10 
may  be  made.  However,  for  the  short  distances 
involved  this  correction  may  generally  be  neglected. 

137.  Consider  that  o  =  \y\,  using  the  subscript 
i  to  designate  the  y  etc  ,  of  the  terminal  point  of 
the  spiral.  By  trigonometry  y\ — o  —  R\  vers  A. 
Then  y\  —  f  A  vers  A  and 

o  =  \  Avers  A (54) 

Also  by  substitution  for  a  and  L  in  equations  (6), 
(7),  (13),  (17),  (21)  and  23  page  20,  the  following 
formulas  are  obtained. 

y  =  A (55) 


v>  —  c      °    nr  c      .  y .  (*&\ 

M\  Z.2  or  A         -[Q     s    '• 


O  I 

fir  si 

o  =  ^-ror 


24  k       24  R!  • 

.' (58) 


c-s~~9oT3 (59) 


=  *5  +  T20^ ( 

In  the  last  three  equations,  note  that  the  t  correc- 
tion is  \  of  the  x  correction  (last  term  in  equation 
(56)  used  in  finding  x  from  5) ;  the  C1  correction  is  |of 
the  x  correction;  and  the  v  correction  is  \  of  the  x 
correction. 


TABLES  83 

For  extreme  cases,  the  values  of  y  and  o  given  by 
equations  (55)  and  (57)  will  be  slightly  too  large. 

57 

For  y,  subtract  .003     3-  from  the  results  of  equation 

K 

(55).  For  o  subtract  one  eighth  as  much.  For  #, 
t,  C  and  z>,  the  terms  given  in  the  equations  above 
will  generally  be  sufficiently  accurate. 

138.  The  following  equations  may  be  repeated 
here  : 

T=  /+  Cffi  +  0)  tani/ (18) 

E=  (jRi  +  0)  exsec  J  /+  o (20) 

u  —  x  —  y  cot  A . .  .  ] 

(22) 

U  —  #  —  27  COS  J J 

139.  Tables. — Tables  will  facilitate  the  applica- 
tion   of   these   equations.     Tables    XV-XIX    give 
properties  of  five  spirals.     The  spiral  is  to  be  used 
up  to  that  length  which  gives  the  required  radius. 
The  x  correction  is  the  amount  to  be  subtracted 
from  the  length  of  spiral  to  give  the  abscissa  x. 
The  long  chord  Cmay  be  found  by  subtracting  four 
ninths  of  the  x  correction  from  the  length  of  spiral. 
Similarily  the  spiral  tangent-distance  v  is  found  by 
adding  one  third  of  the  x  correction  to  one  third  of 
the  length  of  spiral.     Interpolations  for  distances 
between  those  given  in  the  tables  may  be  made,  but 
it  is  best  to  compute  R  and  the  angles. 

140  Laying  out  spiral. — The  same  methods  may 
be  used  in  laying  out  spirals  of  short  radii  as  have 
been  described  for  curves  of  large  radii.  The  loca- 
tion of  the  P.S.,  P.C.C.,  and  P.C.  is  generally  not 
difficult.  If  the  lines  have  been  run  to  an  inter- 


84  STREET  RAILWAY  SPIRALS 

section,  as  is  generally  desirable,  the  tangent-dis- 
tance Z'may  be  measured  to  locate  the  P.S.  The 
P.C.C.  may  be  located  by  turning*  off  the  full  spiral 
deflection  angle  $1  at  the  P.S.  and  measuring  the 
long  chord  C;  or  the  spiral  tangent  distances  u  and 
v  may  be  calculated  and  the  angle  ^i  turned  at  their 
intersection  point.  In  either  case  the  tangent  at 
the  P.C.C.  may  readily  be  found.  Another  method 
is  to  locate  the  P.C.  by  offsetting  the  distance  o 
from  the  initial  tangent  and  then  running  in  the 
circular  curve  to  the  P.C.C. 

141.  Centers   for  the  spiral  may  be  set  (a)   by 
measuring   ordinates   from  the  initial  tangent  for 
the   full  length  of  spiral;    (b)   by  ordinates   from 
the  initial  tangent  as  far  as  the  middle  of  the  spiral 
and  from  the  produced  circular  curve  for  the  re- 
maining  half  length;  (c)   by  ordinates    from   the 
initial-tangent  for  about  two  thirds  of  the  spiral 
and  from   the  terminal  spiral  tangent  for  the  re- 
mainder of  the  spiral.     The  offsets  from  the  circu- 
lar curve  will  be  the  same  for  given  distances  from 
the  P.C.C.  as  thejy  for  an  equal  length  of  spiral. 
The  offsets  from  the   terminal  spiral  tangent  will 
be  the  difference  between  the  offset  for  the  circular 
curve  and  thejy  for  a  spiral,  both  for  a  length  equal 
to  the  distance  from  P.C.C.   to  the  point  located. 

52  58 

This  will  be  -^-j~  —  — -r,  where  5  is  the  distance  of 
2  K       6  k 

the  point  from  the  P.C.C.  and  R  is  the  radius  of 
the  circular  curve. 

142.  Location  by  means  of  deflection  angles 
from  initial  tangent  at  P.S.  is  so  similar  to  that  for 
railway  spirals  already  described  that  it  need  not 


ARC   EXCESS  85 

be  further  discussed.  If  the  rails  have  previously 
been  bent  to  their  proper  curvature  very  few  centers 
need  be  set. 

143.  Arc  excess. — It  must  be  borne  in  mind  that 
the  actual  length  of  arc  is  considered  in  the  formu- 
las here  given,  and  care  must  be   taken   to  provide 
for  the  difference  between  arc  and  chord  measure- 
ment.    The  long1  chord  is  easily  found  by  the  x  cor- 
rection  of  the   tables  as  already   indicated.      For 
other  chords  of  spiral  arcs  (not  from  P.S.)>  it  will 
be  sufficiently  accurate  to  use  for  the  excess  of  arc 
over  chord  the  excess  of  the  same  length  of  circular 
arc  having1   a  radius  equal   to  that  of  the  middle 
point  of  the  spiral  arc  under  consideration.     The 
excess  length  of  a  circular  arc  over  its  chord  may 

(? 
be  calculated  from  the  approximate  formula        ^ 

where  c  may  be  used  as  the  length  of  either  chord 
or  arc.  It  may  also  be  noted  here  that  the  number 

57.3s 
of  degrees  of  angles  in  a  circular  arc  is  — -~ — ,  and 

the  deflection  angle  from  tangent  is  of  course  half 
of  this. 

144.  Curving  rails. — The  principles  here  outlined 
are  for  the  center  line  of  track,  but  it  may  be  de- 
sirable to  have  measurements  for  the  curves  formed 
by  the  rails  for  use  in  curving  rails,  etc.     Although 
the   outer  and   inner   rails   will  be  parallel  to  the 
center  line,  their  lines  will  not  be  true  spirals,  and 
allowance  should  be   made   for   this.     For  data  in 
bending  rails,  it  will  be  well  first  to  get  the  varia- 
tion in  length  of  rail  from  the  length  of  the  center 
line.     For  a  given  point  on  the  center  line,  first  find 


86  STREET   RAILWAY    SPIRALS 

the  d  of  the  spiral.  The  outer  rail  will  be  -^^  ^X^G 

180 

longer  than  the  center  line,  and  the  inner  rail  will 
be  as  much  shorter,  G  being-  the  gauge  of  track. 
Thus  for  k  =  1500  and  a  spiral  distance  of  30  ft., 
^  =  17°  II7  and  the  excess  length  in  outer  rail  is 

17  18 

— ^—  X  3.14  X  2.35  =  .70  ft.     The  outer  rail  dis- 

loU 

tance  will  be  30.70  ft.  and  the  inner  rail  distance 
29.30  ft.  The  ordinate  of  the  rail  from  its  own 
initial  tangent  will  be  the  y  for  the  center  line 
spiral  ±  \G  vers  A  ;  plus  to  be  used  for  the  outer 
rail  and  minus  for  the  inner  one.  Thus  for  the 
example  above  cited,  the  ordinates  for  the  rails  op- 
posite a  center  distant  30  ft.  from  the  P.S.  (30.70 
along  outer  rail  and  29.30  ft.  along  inner  rail)  will 
be  3.00  ±  11  =  3.11  and  2.89.  In  locating  points  on 
the  rails,  allowance  should  be  made  for  the  differ- 
ence between  the  center  line  distance  and  the  rail 
distance.  The  x  for  the  point  on  the  rail  will  be 
the  x  for  the  corresponding  point  on  the  center  line 
spiral  ±  \  6rsin  J.  These  principles  will  apply  to 
any  point  on  the  spiral.  It  will  be  well  to  tabulate 
values  for  the  sets  of  curves  most  used. 

145  If  it  is  desired  to  locate  the  last  third  of 
the  spiral  from  the  terminal  spiral  tangent  for  the 
two  rails,  the  length  and  position  of  these  tangents 
may  be  calculated  from  their  ordinates  and  the  ^ 
already  used,  and  points  on  the  rails  located  by  off- 
sets from  these  tangents.  These  offsets  may 
readily  be  calculated  by  the  principles  already  out- 
lined. 


DOUBLE  TRACK  87 

146.  Double  track. — Double  track  curves  will 
generally  need  radii  of  different  lengths.  The 
spirals  for  both  curves  may  be  taken  from  the  same 
table,  or  the  one  for  the  outside  curve  may  be  taken 
from  the  table  having-  a  value  of  k  next  higher  than 
that  used  with  the  inside  curve.  In  the  latter  case 
the  two  spirals  will  be  of  nearly  the  same  length  and 
their  ends  will  be  nearly  opposite.  If  the  distance 
between  center  lines  on  the  curve  be  made  equal  to 
the  distance  between  center  lines  on  tangent  plus 
the  difference  in  the  o  for  the  inside  and  outside 
spiral,  the  circular  parts  of  the  two  curves  will  be 
parallel  and  have  the  same  center,  and  the  radius  of 
the  outside  curve  will  be  equal  to  the  radius  of  the 
inside  curve  plus  the  distance  between  tracks  on  the 
curve. 

1 47-  In  case  consideration  of  clearance  requires 
greater  distance  between  the  tracks  on  curves  than 
on  tangents,  care  must  be  exercised  in  the  selection 
of  the  spiral.  The  calculation  of  the  external-dis- 
tance E  will  best  enable  the  distance  between  center 
lines  at  their  middle  points  to  be  determined.  If  the 
inside  radius  is  assumed,  first  find  its  external  dis- 
tance E,  to  this  add  the  distance  from  one  P.  I.  to 
the  other,  and  subtract  the  required  distance  be- 
tween the  curves.  The  remainder  will  be  the  ex- 
ternal-distance E  for  the  outside  curve,  from  which 
the  desired  radius  may  be  found.  As  by  this  ar- 
rangement the  two  curves  will  be  closer  at  their 
ends  than  at  their  middle,  care  must  be  taken  to 
secure  sufficient  clearance.  The  selection  of  curves 
and  spirals  for  double  track  is  more  complex  than 
for  single  track. 


88  CONCLUSION 

CONCLUSION. 

148.  Besides  the   problems   and   methods   here 
presented,  many  other  applications  may  be  made. 
For  particular  conditions  the  engineer  may  develop 
special  methods. 

The  preceding-  methods  g-enerally  have  been  based 
-upon  the  principle  that  the  spiral  is  to  have  the 
same  degree-of-curve  at  the  end  as  the  main  curve, 
and  slig-ht  modifications  may  be  necessary  when 
not  so.  The  value  of  o  and  of  the  angle  in  the 
circular  curve  omitted  must  be  that  for  the  spiral 
used.  Thus  with  a  —  2  the  spiral  at  the  end  of  300 
ft.  will  be  a  6°  curve.  It  may,  however,  be  there 
compounded  with  a  curve  of  different  radius,  as  a  6° 
30 '  curve,  provided  the  offset  is  3.93  and  the  cen- 
tral angle  between  the  P.C.  and  the  P.C.C.  is  9°. 
Generally,  in  order  to  utilize  the  problems  given  for 
old  track,  etc.,  the  formulas  will  need  to  be  modified 
if  Z^o  does  not  agree  with  D\. 

149.  In  field  work  most  of  the  usual  formulas  of 
of  the  various  location  problems,  like  "Required  to 
chang-e  the  P.C.  so  that  the  curve  may  end  in  a  par- 
allel tang-ent,"  may  be  used  without  modification 
with   curves  having  transition  ending's,  by  simply 
considering-  the  whole  intersection  angle  including- 
the  angle  in  the  spirals.     This  is  true  whenever  the 
same  amount  of  spiral  is  used  with  the  new  curve. 

If  the  degree-of-curve  chang-es  and  with  it  the 
leng-th  of  the  spiral,  the  difference  between  the  0's 
in  the  two  cases  must  be  allowed  for.  With  a  little 
practice  in  using-  such  formulas  with  spirals,  the 
engineer  will  find  no  difficulty. 


CONCLUSION  89 

1  50.  The  transition  spiral  has  the  merit  of  com- 
parative simplicity  and  extreme  flexibility.  It  is  a 
natural  method,  since  it  is  so  similar  to  the  methods 
used  in  laying*  out  circular  curves.  Like  circular 
curves,  the  length  along-  the  curve  is  the  principal 
term,  and  the  degree-of-curve,  central  ang-le,  de- 
flection angles  and  ordinates  are  obtainable  from 
this  variable.  It  may  be  used  with  any  main  curve, 
even  of  fractional  degree  ;  any  leng-th  of  chord  may 
be  used  in  measurement  under  the  same  restrictions 
as  circular  curves,  and  as  it  is  not  necessary  to  re- 
strict the  measurements  to  a  common  chord  leng-th, 
intermediate  points  may  be  readily  located.  The 
calculations  for  angles  and  distances  are  easily 
made.  If  desired,  the  tangent  and  the  circular 
curve  may  be  run  out  and  the  spiral  put  in  by  co- 
ordinates, one  half  from  the  tang-ent  and  one  half 
from  the  circular  curve.  This  is  especially  appli- 
cable to  location  work  and  to  short  spirals. 

The  engineer  should  not  be  frig-htened  by  the 
mathematics  in  the  demonstration  of  the  formulas ; 
the  principles  and  methods  may  be  understood  with- 
out mastering-  the  demonstrations.  Experience  has 
shown  that  the  ordinary  transit  man,  with  a  little 
thought  and  study,  can  understand  and  use  the 
transition  spiral  as  easily  as  circular  curves,  and  that 
young-  assistants  without  previous  training-  readily 
take  up  the  work. 

151.  With  reference  to  the  use  of  the  cubic  para- 
bola as  an  easement  it  may  be  said  that,  except 
for  the  relation  between  oc  andy,  it  has  no  proper- 
ties of  value  for  a  transition  curve  which  are  not 
merely  approximations  of  the  transition  spiral. 


90  CONCLUSION 

Within  small  limits,  the  radius  of  curvature  and 
the  angles  to  be  used  approach  somewhat  closely  to 
those  for  the  transition  spiral.  As  soon  as  oc  differs 
materially  from  the  length  of  curve,  a  correction 
has  to  be  made.  The  radius  of  curvature  finally 
begins  to  increase.  The  investigation  of  the  cubic 
parabola  in  reference  to  its  radius  of  curvature,  its 
angle  turned,  the  angular  deflection  to  points  on  it, 
and  the  length  of  the  curve,  require  as  long  mathe- 
matical equations  as  those  governing  the  transition 
spiral.  Many  attempts  have  been  made  to  utilize 
this  curve,  but  both  field  work  and  computations 
are  too  intricate  and  inconvenient  if  the  curve  has 
any  considerable  length,  and  it  has  no  advantage 
over  the  transition  spiral. 

152.  The  question  of  the  efficiency  of  easement 
curve  is  of  considerable  importance.  The  objection 
is  sometimes  raised  that  even  if  track  is  laid  out 
with  a  carefully  fitted  spiral  there  would  be  no  pos- 
sibility of  keeping  it  in  place  by  the  methods  of  the 
ordinary  trackman.  This  identical  objection  could 
be  made  with  the  same  force  against  carefully  laid 
out  circular  curves,  yet  no  engineer  would  recom- 
mend abolishing  that  practice.  Even  if,  in  re-lin- 
ing, the  transition  curve  is  considerably  distorted^ 
it  remains  an  easement,  and  will  be  in  far  better 
riding  condition  than  a  distorted  circular  curve. 
By  marking  the  P.S.  and  the  P.C.C.  with  a  stake 
or  post,  with  intermediate  points  on  long  spirals, 
the  trackman  will  be  able  to  keep  the  spiral  in  as 
good  condition  as  though  it  were  of  uniform  curva- 
ture. The  short  spirals  advocated  by  some  engi- 
neers have  proved  to  be  insufficient.  For  efficient 


CONCLUSION  91 

service,  a  length  of  spiral  which  will  give  an  o  of 
considerable  amount  must  be  used,  even  if  this 
necessitates  widening-  the  roadbed, 

153.  Properly  constructed  spirals  would  fre- 
quently allow  the  use  of  sharper  curvature — since 
the  riding-  quality  of  curves  may  be  the  governing 
consideration  in  the  selection  of  a  maximum  —  and 
thus  make  a  saving-  in  construction.  By  fitting- 
curves  with  proper  transition  spirals,  roads  using* 
sharp  curves  may  partially  relieve  the  objection 
of  the  public  to  traveling-  by  their  routes.  The 
introduction  of  fast  trains  has  made  it  necessary  to 
take  every  precaution  to  secure  an  easy-riding  track. 
The  disagreeable  lurch  and  necessary  "  slow  order  " 
for  fast  trains  at  certain  curves  on  man}7  roads  has 
been  entirely  eliminated  by  the  construction  of 
proper  spirals,  and  passengers  do  not  now  know 
when  such  curves  are  reached.  The  transition 
curve  has,  then,  a  financial  value  largely  overbal- 
ancing its  cost.  The  adoption  of  such  curves  by 
many  of  our  principal  railways  proves  their  effici- 
ency, and  the  futnre  will  see  a  much  more  general 
adoption. 


Q2  EXPLANATION  OF  TABLES 


EXPLANATION  OF  TABLES. 

In  tables  I-XI,  the  columns   g-ive  the  following- 
properties  : 

1.  The  distance  in  feet  from  the  P.S.  along- the 
spiral  to  a  point  on  the  spiral  ;  i.  e.  100  L.     The  full 
leng-th  of  spiral  will  g-ive  values  for  the  terminal 
point,  the  P.C.C.  of  main  curve. 

2.  D,  the  degree-of-cur ve  of  the  spiral  at  any  point. 
It  becomes  Di  at  the  P.C.C. 

3.  4  the  spiral  ang-le  or  chang-e  of  direction  of  the 
spiral  to  the  point. 

„ £ ^ 


P.S  ' 


•R\C. 


„  >\p.c.c. 


FIG.  17 

4.  9,  the  spiral  deflection  ang-le  at  the  P.S.  from 
the  initial  tang-ent  to  locate  the  point. 

5.  o,  the  offset  from  the  initial  tang-ent  to  the  P. 
C.  of  main  curve  produced   backward.     Enter  the 
table  with  the  full  leng-th  of  the  spiral  used. 

6.  y,  the  ordinate  from  the  initial  tang-ent  as  the 
axis  of  X. 

7.  x  COR.,  an  amount  to  be  subtracted  from  the 
distance  in  feet  from  the  P.S.  along-  the  spiral  to 
find  the  abscissa,  #,  of  the  point,     x  —  100  L  -#CoR. 


EXPLANATION  OF  TABLES  93 

8.  /CoR.,  an  amount  to  be  subtracted  from  half 
the  full  leng-th  of  the  spiral  in  feet  to  find  /,  the 
abscissa  of  the  P.C.  Enter  the  table  with  full 
leng-th  of  the  spiral  used,  t  =  ±%°  L  —t  COR. 

To  find  the  long-  chord  to  P.S.,  subtract  four 
ninths  (.444)  of  x  COR.  from  the  leng-th  of  the  spiral 
in  feet.  C=100  L  —  %x  COR.  For  chords  not  end- 
ing- at  P.S.,  see  pag-e  85. 

To  find  the  terminal  spiral  tang-ent-distance,  add 
one  third  x  COR.  to  one  third  the  spiral  distance  to 
the  point.  z>  =  i{p  L  -+  \  x  COR. 

Intermediate  values  may  be  found  by  interpola- 
tion. 

With  transit  at  intermediate  point  on  spiral,  for 
deflection  ang-le  0,  see  pag-es  11,  23,  and  44. 

To  use  Table  IV  for  other  values  of  a,  multiply 
the  tabulated  values  of  D,  A,  9,  o,  andjy  in  Table 
IV  opposite  the  given  distance  from  the  P.S.  by  the 
a  of  the  desired  spiral,  and  x  COR.  and  /  COR.  by  the 
square  of  a.  For  inaccuracies  of  this  method  see 
pag-e  31.  If  a  and  D  or  o  are  given,  first  find  L. 

Table  XII  permits  ordinates  to  be  calculated  from 
o.  See  Fig-.  5. 

For  the  use  of  Tables  XIII  and  XIV  see  pag-e  74, 
and  for  Tables  XV-XIX,  see  pag-e  83. 

Table  XX  gives  values  of  o  and  L  for  values  of  a 
and  D. 

For  full  nomenclature,  see  pag-e  3. 

For  equations  and  summary  of  principles,  see  pag-e 

20. 

For  fuller  explanation  of  tables  and  errors  of  in- 
terpolation, see  pag-e  28. 

For  choice  of  #,  see  pag-e  32. 


1°  in  200  ft. 


TABLE  I.     TRANSITION  SPIRAL. 


1 

Length 

D 

A 

e 

o 

y 

x  COR. 

/COR. 

25 

Q0W%' 

o°oo:9 

o°oo:3 

.00 

.00 

50 

0  15 

0  03.8 

0  01.3 

.00 

.02 

75 

0  22Y2 

0  08.4 

0  02.8 

.02 

.06 

100 

0  30 

0  15. 

0  05. 

.04 

.15 

125 

0  37^ 

0  23.4 

0  07  8 

.07 

.29 

150 

0  45 

0  33.8 

0  11.3 

.12 

.49 

175 

0  52^ 

0  45.9 

0  15.3 

.20 

.78 

.00 

200 

1  00 

1  00. 

0  20. 

.29 

1.16 

.01 

225 

1  07^ 

1  15.9 

0  25.3 

.41 

1.66 

.01 

250 

I  15 

1  33.8 

0  31.3 

.57 

2.27 

.02 

.00 

275 

1  22^ 

1  53.4 

0  37.8 

.76 

3.03 

.03 

.01 

300 

1  30 

2  15. 

0  45. 

.98 

3.93 

.05 

.01 

325 

1  37^ 

2  38.4 

0  52.8 

1.25 

5.00 

.07 

.01 

350 

1  45 

3  03.8 

1  01.3 

1.56 

6.23 

.10 

.02 

375 

1  52^ 

3  30.9 

1  10.3 

1.92 

7.67 

.14 

.02 

400 

2  00 

4  00. 

1  20. 

2.33 

9.31 

.19 

.03 

425 

2  07^ 

4  30.9 

1  30.3 

2.79 

11.16 

.26 

.04 

450 

2  15 

5  03.8 

1  41.3 

3.31 

13.25 

.35 

.06 

475 

2  22% 

5  38.4 

1  52.8 

3.89 

15.58 

.46 

.08 

500 

2  30 

6  15. 

2  05. 

4.54 

18.16 

.59 

.10 

525 

2  37^ 

6  53.4" 

2  17.8 

5.26 

21.03 

.75 

.13 

550 

2  45 

7  33.8 

2  31.3 

6.04 

24.17 

.95 

.16 

575 

2  52/2 

8  15.9 

2  45.3 

6.91 

27.62 

1.20 

.20 

600 

3  00 

9  00. 

3  00. 

7.84 

31.36 

1.48 

.24 

625 

3  07^ 

9  45.9 

3  15.3 

8.87 

35.45 

1.81 

.30 

650 

3  15 

10  33.8 

3  31.3 

9.97 

39.85 

2.21 

.37 

675 

3  22y2 

11  23.4 

3  47.8 

11.16 

44.63 

2.66 

.44 

700 

3  30 

12  15. 

4  04.9 

12.45 

49.73 

3.20 

.53 

725 

3  37^ 

13  08.4 

4  22.7 

13.83 

55.22 

3.81 

.64 

750 

3  45 

14  03.8 

4  41.2 

15.30 

61.09 

4.51 

.75 

775 

3.52^ 

15  00.9 

5  00.1 

16.88 

67.37 

5.31 

.89 

800 

4  00 

16  00. 

5  19.8 

18.56 

74.05 

6  22 

1  01 

1°  in  150  ft. 


TABLK  II.     TRANSITION  SPIRAL. 


Length 

D 

J 

9 

0 

y 

x  COR. 

/fCOR. 

25 

0°10' 

0001'3 

0°00'4 

.00 

.00 

50 

0  20 

0  05. 

0  01.7 

.01 

.02 

75 

0  30 

0  11.3 

0  03.8 

.02 

.08 

100 

0  40 

0  20. 

0  06.7 

.05 

.19 

125 

0  50 

0  31.3 

0  10.4 

,10 

.38 

150 

1  00 

0  45. 

0  15. 

.16 

.65 

0.00 

175 

1  10 

1  01.3 

0  20.4 

.26 

1.04 

0.01 

200 

1  20 

1  20. 

0  26.7 

.39 

1  55 

0  01 

TABLE  II.— Continued. 


1°  in  150  ft. 


Length 

D 

A 

e 

0 

y 

x  COR. 

/COR. 

225 

1°30' 

1°41'.3 

0°33'.8 

.55 

2.21 

.02 

.00 

250 

1  40 

2  05. 

0  41.7 

.76 

3  03 

.03 

.01 

275 

1  50 

2  31.3 

0  50.4 

1  01 

4.04 

.05 

.01 

300 

2  00 

3  00. 

1  00. 

1.31 

5.23 

.08 

.01 

325 

2  10 

3  31.3 

1  10.4 

1.66 

6.66 

.12 

.02 

350 

2  20 

4  05. 

1  21.7 

2.08 

8.31 

.18 

.03 

375 

2  30 

4  41.3 

1  33.8 

2.56 

10  23 

.25 

.04 

400 

2  40 

5  20. 

1  46.7 

3.10 

12.40 

.35 

.06 

425 

2  50 

6  01.3 

2  00.4 

3.72 

14  88 

.47 

.08 

450 

3  00 

6  45. 

2  15. 

4.41 

17.66 

.62 

.10 

475 

3  10 

7  31.3 

2  30.4 

5.19 

20  76 

.82 

.14 

500 

3  20 

8  20. 

2  46.7 

6.05 

24.20 

1.06 

.18 

525 

3  30 

9  11.3 

3  03.8 

7.01 

28.02 

1.35 

.22 

550 

3  40 

10  05. 

3  21.7 

8.05 

32  19 

1.70 

.28 

575 

3  50 

11  01.3 

3  40.4 

9.20 

36.78 

2.12 

.36 

600 

4  00 

12  00 

3  59.9 

10.45 

41  76     | 

2.63 

.44 

TABLE  III.     TRANSITION  SPIRAL. 


1°  in  125  ft. 


Length 

D 

A 

8 

0 

y 

x  COR. 

*COR. 

25 

0°12' 

0C01^' 

0°00>^  ' 

.00 

.oT~ 

50 

0  24 

0  06 

0  02 

.01 

.03 

75 

0  36 

0  13  # 

004/2 

.02 

.10 

100 

0  48 

0  24 

0  08 

.06 

.23 

125 

1  00 

0  37^ 

0  12# 

.11 

.46 

150 

1  12 

0  54 

0  18 

.20 

.79 

.00 

175 

1  24 

1  13# 

0  24/2 

.31 

1.25 

.01 

200 

1  36 

1  36 

0  32 

.47 

1.86 

.02 

225 

1  48 

2  Ol/2 

0  40^ 

.66 

2.65 

.03 

.00 

250 

2  00 

2  30 

0  50 

.91 

3.64 

.05 

.01 

275 

2  12 

3  01X 

1  00>^ 

1.21 

4.84 

.08 

.01 

300 

2  24 

3  36 

1  12 

1.57 

6.28 

.12 

.02 

325 

2  36 

4  13# 

1  24^ 

2.00 

7.99 

.18 

.03 

350 

2  48 

4  54 

1  38 

2.49 

9.97 

.26 

.04 

375 

3  00 

5  37^ 

1  52K 

3.07 

12.27 

.36 

.06 

400 

3  12 

6  24 

2  08 

3.72 

14.88 

.50 

.08 

425 

3  24 

7  13K 

2  24^ 

4.47 

17.85 

.68 

.11 

450 

3  36 

8  06 

2  42 

5.31 

21.18 

.90 

.15 

475 

3  48 

9  01# 

3  00^ 

6.23 

24.90 

1.18 

.20 

500 

4  00 

10  00 

3  20 

7.26 

29.02 

1.52 

.25 

IV.     TRANSITION  SPIRAL, 


1°  in  100  ft. 


Length 

D 

J 

0 

0 

y 

x  COR. 

/CoR. 

10 

0.1° 

o°oo:3 

o°oo:i 

.000 

"Tooo" 

.000 

.000 

20 

0.2 

01.2 

00.4 

.001 

.002 

30 

0.3 

02.7 

00.9 

.002 

.008 

40 

0.4 

04.8 

01.6 

.005 

.019 

50 

0.5 

07.5 

02.5 

.009 

.036 

60 

0.6 

0  10.8 

0  03.6 

.016 

.063 

70 

0.7 

14.7 

04.9 

.025 

.100 

80 

0.8 

19.2 

06.4 

.037 

.149 

90 

0.9 

24.3 

08.1 

053 

.212 

100 

1.0 

30. 

10. 

.073 

.291 

.001 

110 

1.1 

0  36.3 

0  12.1 

.097 

.387 

.001 

120 

1.2 

43.2 

14.4 

.126 

.503 

.002 

130 

1.3 

50.7 

16.9 

.160 

.639 

.003 

140 

1.4 

58.8 

19.6 

.199 

.798 

.004 

150 

1.5 

1  07.5 

22.5 

.245 

.982 

.006 

.001 

160 

1.6 

1  16.8 

0  25.6 

.298 

1.191 

.008 

.001 

170 

1.7 

1  26.7 

28.9 

.357 

1.429 

.011 

.002 

180 

1.8 

1  37.2 

32.4 

.424 

1.696 

.014 

.002 

190 

1.9 

1  48.3 

36.1 

.499 

1.995 

.019 

.003 

200 

2.0 

2  00. 

40. 

.582 

2.327 

.024 

.004 

-210 

2.1 

2  12.3 

0  44.1 

.673 

2.690 

.031 

.005 

220 

2  2 

2  25.2 

48  4 

.774 

3.097 

.039 

.006 

230 

2.3 

2  38.7 

52.9 

.885 

3.538 

.049 

.008 

240 

2  4 

2  52.8 

57.6 

1.005 

4.020 

.061 

.010 

250 

2.5 

3  07.5 

1  02.5 

1.136 

4.544 

.074 

.012 

260 

2.6 

3  22.8 

1  07.6 

1.278 

5.111 

.090 

.015 

270 

2.7 

3  38.7 

1  12.9 

1.431 

5.724 

.109 

.018 

280 

2.8 

3  55.2 

1  18.4 

1.596 

6.383 

.131 

.022 

290 

2.9 

4  12.3 

1  24.1 

1.773 

7.091 

.156 

.027 

300 

3.0 

4  30. 

1  30. 

1.963 

7.850 

.185 

.031 

310 

3.1 

4  48.3 

1  36.1 

2.166 

8.66 

.218 

.036 

320 

3.2 

5  07.2 

1  42.4 

2.382 

9.53 

.255 

.043 

330 

3.3 

5  26.7 

1  48.9 

2.612 

10.45 

.298 

.050 

340 

3.4 

5  46.8 

1  55.6 

2.857 

11.42 

.346 

.058 

350 

3.5 

6  07.5 

2  02.5 

3.116 

12.46 

.400 

.067 

360 

3.6 

6  28.8 

2  09.6 

3.391 

13.56 

.460 

.077 

370 

3.7 

6  50.7 

2  16.9 

3.681 

14.72 

.528 

.088 

380 

3.8 

7  13  2 

2  24.4 

3.988 

15.94 

.603 

.100 

390 

3.9 

7  36.3 

2  32.1 

4.311 

17.23 

.686 

.114 

400 

4.0 

8  00. 

2  40. 

4.651 

18.59 

.779 

.130 

410 

4.1 

8  24.3 

2  48.1 

5.01 

20.02 

.881 

.147 

420 

4.2 

8  49.2 

2  56.4 

5.38 

21.51 

.994 

.166 

430 

43 

9  14.7 

3  04.9 

5.78 

23.08 

1.118 

.186 

440 

4.4 

9  40.8 

3  13.6 

6.19 

24.73 

1.254 

.209 

450 

4.5 

10  07.5 

3  22.5 

6.62 

26.45 

1.403 

.234 

1°  in  100  ft. 


IV.— Continued. 


0=1. 


Length 

D 

A 

0 

0 

y 

*:COR. 

/CoR. 

460 

4.6° 

10°34:8 

3°31'.6 

7.07 

28  24 

1.57 

.26 

470 

4.7 

11  02.7 

3  40.9 

7.54 

30.12 

1.74 

.29 

480 

4.8 

11  31.2 

3  50.4 

8  03 

32.07 

1.94 

.32 

490 

4.9 

12  00.3 

4  00.1 

8.54 

34.11 

2.15 

.36 

500 

5.0 

12  30. 

4  10. 

9  07 

36.23 

2.37 

.40 

510 

5.1 

13  00.3 

4  20.1 

9.63 

38.44 

2.62 

.44 

520 

5.2 

13  31  2 

4  30.4 

10.20 

40.73 

2.89 

.48 

530 

53 

14  02.7 

4  40.9 

10.80 

43.12 

3.17 

.53 

540 

5.4 

14  34.8 

4  51.4 

11.42 

45.59 

3.49 

.58 

550 

5.5 

15  07.5 

5  02.3 

12.07 

48.15 

3.82 

.64 

560 

5.6 

15  40.8 

5  13.4 

12.74 

50.83 

4.18 

.70 

570 

5.7 

16  14.7 

5  24.7 

13.43 

53.56 

4.56 

.76 

580 

5.8 

16  49.2 

5  36.2 

14.14 

56.40 

4.98 

.83 

590 

5.9 

17  24.3 

5  47.8 

14.89 

59.34 

5.42 

.90 

600 

6.0 

18  00. 

5  59.7 

15.65 

62.39 

5.89 

.98 

610 

6.1 

18  36.3 

6  11.8 

16.44 

65.52 

6.40 

1.07 

620 

6.2 

19  13.2 

6  24.1 

17.26 

68.77 

6.94 

1.16 

630 

6.3 

19  50.7 

6  36.5 

18.10 

72.11 

7.51 

1  25 

640 

6.4 

20  28.8 

6  49.1 

18.97 

75.56 

8.13 

1.36 

650 

6.5 

21  07.5 

7  02.0 

19.87 

79.11 

8.78 

1.47 

660 

6.6 

21  47. 

7  15.1 

20.79 

82.79 

9.48 

1.57 

670 

6.7 

22  27. 

7  28.5 

21.74 

86.56 

10.22 

1.69 

680 

6.8 

23  07. 

7  41.8 

22.73 

90.43 

11.00 

1.82 

690 

6.9 

23  48. 

7  55.3 

23.73 

94.42 

11.82 

1.96 

700 

7.0 

24  30. 

8  09.3 

24.79 

98.50 

12  70 

2.10 

TABLE  V.     TRANSITION  SPIRAL. 


in  80  ft. 


Length 

D 

A 

0 

o 

y 

x  COR. 

/COR. 

10 

0°07i' 

o°ootf 

0°00' 

.00 

.00 

.00 

.00 

20 

0  15 

0  Oli 

o  001 

.00 

.00 

30 

0  221 

0  03^ 

0  01 

.00 

.01 

40 

0  30 

0  06 

0  02 

.00 

.02 

50 

0  371 

0  091 

0  03 

.01 

.04 

60 

0  45 

0  131 

0  041 

.02 

.08 

70 

0  521 

0  181 

0  06 

.03 

.12 

80 

1  00 

0  24 

0  08 

.05 

.19 

90 

1  07J- 

0  301 

0  10 

.07 

.26 

100 

1  15 

0  371 

0  121 

.09 

.36 

110 

1  221 

0  45ls 

0  15 

.12 

.48 

120 

1  30 

0  54 

0  18 

..16 

.63 

130 

1  371 

1  031 

0  21 

.20 

.80 

140 

1  45 

1  131 

0  24J 

.25 

1.00 

150 

1  521 

1  241 

0  28 

.31 

1.23 

.00 

.00 

1°  in  80  ft. 


TABLE  V.— -Continued. 


Length 

'D 

4 

9 

0 

y 

x  COR. 

/'COR. 

160 

2°00' 

1°36' 

0°32' 

.37 

1.49 

~."o~" 

~7o~ 

170 

2  071 

1  48* 

0  36 

.45 

1.77 

180 

2  15 

2  01* 

0  40* 

.53 

2.12 

190 

2  22* 

2  15* 

0  45 

.62 

2.50 

200 

2  30 

2  30 

0  50 

.73 

2.90 

210 

2  37* 

2  45* 

0  55 

.84 

3.36 

220 

2  45 

3  01* 

1  00* 

.97 

3.87 

230 

2  521 

3  18* 

1  06 

1.10 

4.42 

240 

3  00 

3  36 

1  12 

1.25 

5.02 

250 

3  071 

3  54* 

1  18 

1.42 

5.67 

.1 

260 

3  15 

4  13* 

1  24* 

1.59 

6.38 

.1 

270 

3  22* 

4  331 

1  31 

1.79 

7.15 

.2 

280 

3  30 

4  54 

1  38 

1.99 

7.98 

.2 

290 

3  37* 

5  151 

1  45 

2.21 

8.86 

.2 

300 

3  45 

5  371 

1  52* 

2.45 

9.81 

.3 

310 

3  521 

6  00* 

2  00 

2.70 

10.74 

.3 

320 

4  00 

6  24 

2  08 

2.98 

11.91 

.4 

330 

4  07* 

6  48* 

2  16 

3.26 

13.06 

.4 

340 

4  15 

7  13* 

2  24* 

3.57 

14.28 

.5 

350 

4  22* 

7  39* 

2  33 

3.89 

15.57 

.6 

360 

4  30 

8  06 

2  42 

4.23 

16.95 

.7 

.1 

370 

4  37* 

8  33J 

2  51 

4.59 

18.40 

.8 

.1 

380 

4  45 

9  01* 

3  00* 

4.97 

19.92 

.9 

.2 

390 

4  52* 

9  30* 

3  10 

5.38 

21.54 

1.0 

.2 

400 

5  00 

10  00 

3  20 

5.80 

23.23 

1.2 

.2 

410 

5  07* 

10  30* 

3  30 

6.26 

25.00 

1.4 

.2 

420 

5  15 

11  01* 

3  40* 

6.72 

26.86 

1.6 

.3 

430 

5  22J 

11  33* 

3  51 

7.22 

28.82 

1.7 

.3 

440 

5  30 

12  06 

4  02 

7.74 

30.87 

2.0 

.3 

450 

5  37* 

12  39* 

4  13 

8.28 

33.02 

2.2 

.4 

460 

5  45 

13  13* 

4  24* 

8.84 

35.25 

2.4 

.4 

470 

5  52* 

13  48* 

4  36 

9.41 

37.59 

2.7 

.5 

480 

6  00 

14  24 

4  48 

10.03 

40.02 

3.0 

.5 

4913 

6  07i 

15  00* 

5  00 

10.67 

42.56 

3.4 

.6 

500 

6  15 

15  37* 

5  12* 

11.33 

45.20 

3.7 

.6 

510 

6  22* 

16  15J 

5  25 

12.03 

47.95 

4.1 

.7 

520 

6  30 

16  54 

5  38 

12.74 

50.79 

4.5 

.8 

530 

6  37* 

17  33* 

5  51 

13.48 

53.76 

5.0 

.8 

540 

6  45 

18  13* 

6  04 

14.26 

56.84 

5.4 

.9 

550 

6  52* 

18  541 

6  18 

15.07 

60.02 

6.0 

1.0 

560 

7  00 

19  36 

6  32 

15.90 

63  34 

6.5 

1.1 

570 

7  071 

20  18* 

6  46 

16.76 

66.72 

7.1 

1.2 

580 

7  15 

21  01* 

7  00 

17.65 

70.26 

7.8 

1.3 

590 

7  22* 

21  45* 

7  14* 

18.57 

73.90 

8.4 

1.4 

600 

7  30 

22  30 

7  29 

19.52 

77  68 

9.2 

1  5 

TABLK  VI.— TRANSITION  SPIRAL 


1°  in  60  ft. 


Length 

D 

A 

9 

S^T 

•aHfcs^ 

^COR. 

/COR. 

10 

0°10' 

0°00*' 

0°00' 

.00 

.00 

.0 

.0 

20 

0  20 

0  02 

0  001 

30 

0  30 

0  041 

0  OH 

40 

0  40 

0  08 

0  03 

.03 

50 

0  50 

o  12* 

0  04 

.06 

60 

1  00 

0  18 

0  06 

.03 

.10 

70 

1  10 

0  24* 

0  08 

.04 

.17 

80 

1  20 

0  32 

0  10* 

.06 

.25 

90 

1  30 

0  40* 

o  13* 

.09 

.35 

100 

1  40 

0  50 

0  161 

.12 

.48 

110 

1  50 

i  oo* 

0  20 

.16 

.64 

120 

2  00 

1  12 

0  24 

.21 

.84 

130 

2  10 

1  241 

0  28 

.26 

1.06 

140 

2  20 

1  38 

0  32* 

.33 

1.33 

150 

2  30 

1  52* 

0  37* 

.41 

1.63 

160 

2  40 

2  08 

0  42* 

.50 

1.98 

170 

2  50 

2  24* 

0  48 

.59 

2.38 

180 

3  00 

2  42 

0  54 

.70 

2.82 

190 

3  10 

3  00* 

1  00 

.83 

3  32 

200 

3  20 

3  20 

1  06* 

.97 

3.88 

210 
220 

3  30 
3  40 

3  40i 

4  02 

1  13} 

1  201 

1.12 
1.29 

4.48 
5.15 

.1 
.1 

230 

3  50 

4  241 

1  28 

-1.47 

5.90 

.1 

240 

4  00 

4  48 

1  36 

1.67 

6.69 

.2 

250 

4  10 

5  121 

1  44 

1.89 

7.58 

.2 

260 

4  20 

5  38 

1  52k 

2.13 

8.52 

.2 

270 

4  30 

6  04* 

2  OH 

2.38 

9.54 

.3 

280 

4  40 

6  32 

2  10  J- 

2.65 

10.64 

.4 

290 

4  50 

7  00* 

2  20 

2.94 

11.82 

.4 

300 

5  00 

7  30 

2  30 

3.26 

13.07 

.5 

310 

5  10 

8  00* 

2  40 

3.60 

14.43 

.6 

.1 

320 

5  20 

8  32 

2  50* 

3.96 

15.87 

.7 

.1 

330 

5  30 

9  041 

3  OH 

4.34 

17.40 

.8 

.1 

340 

5  40 

9  38 

3  12* 

4.75 

19.02 

.9 

.2 

350 

5  50 

10  121 

3  24 

5.18 

20.74 

1.1 

.2 

360 

6  00 

10  48 

3  36 

5.64 

22.56 

1.3 

.2 

370 

6  10 

11  24* 

3  48 

6.12 

24.50 

1.4 

.2 

380 

6  20 

12  02 

4  00* 

6.63 

26.53 

1.7 

.3 

390 

6  30 

12  401 

4  13* 

7.16 

28.67 

1.9 

.3 

400 

6  40 

13  20 

4  26* 

7.73 

30.92 

2.2 

.4 

410 

6  50 

14  00* 

4  40 

8  34 

'33.27 

2.4 

.4 

420 

7  00 

14  42 

4  54 

8.96 

35.73 

2.8 

.5 

430 

7  10 

15  24* 

5  08 

9.61 

38.32 

3.1 

.5 

440 

7  20 

16  08 

5  22* 

10.30 

41.07 

3.5 

.6 

450 

7  30 

16  521 

5  37* 

11.01 

43.90 

3.9 

.6 

1°  in  60  ft. 


TABLE)  VI.— Continued. 


a=l%. 


length 

D 

J 

e 

0 

y 

x  COR. 

/COR. 

460 

7°40' 

17°38' 

5°52' 

11.75 

46.86 

4.3 

.7 

470 

7  50 

18  24i 

6  08 

12.50 

49.94 

4.8 

.8 

480 

8  00 

19  12 

6  24 

13.35 

53.16 

54 

.9 

490 

8  10 

20  OOi 

6  40 

14.19 

56.52 

5.9 

1.0 

500 

8  20 

20  50 

6  56 

15.07 

60.01 

6.6 

1.1 

510 

8  30 

21  40i 

7  13 

16.00 

63.64 

7.2 

1.2 

520 

8  40 

22  32 

7  30 

16.94 

67.36 

8.0 

1.3 

530 

8  50 

23  24i 

7  471 

17.93 

71.25 

8.8 

1.5 

540 

9  00 

24  18 

8  05 

18.95 

75.31 

9.6 

1.6 

550 

9  10 

25  12i 

8  23 

20.03 

79.53 

10.5 

1.8 

560 

9  20 

26  08 

8  42 

21.13 

83.88 

11.5 

1.9 

570 

9  30 

27  04i 

9  00£ 

22.26 

88.31 

12.6 

2.1 

580 

9  40 

28  02 

9  19i 

23.42 

92.92 

13.7 

2.3 

590 

9  50 

29  00* 

9  39 

24.67 

97.70 

14.9 

2.5 

600 

10  00 

30  00 

9  59 

25.91 

102.66 

16.2 

2.7 

TABLE  VII.     TRANSITION  SPIRAL. 
1°  in  50  ft. 


Length 

D 

A 

0 

0 

y 

x  COR. 

/COR. 

10  . 

0°12' 

0°00i' 

0°00' 

.00 

.00 

.0 

.0 

20 

0  24 

0  02£ 

0  01 

30 

0  36 

0  051 

0  02 

.02 

40 

0  48 

0  09i 

0  03 

.01 

.04 

50 

1  00 

0  15 

0  05 

.02 

.07 

60 

1  12 

0  2H 

0  07 

.03 

.13 

70 

1  24 

0  29£ 

0  10 

.05 

.20 

80 

1  36 

0  38i 

0  13 

.07 

.30 

90 

1  48 

0  48i 

0  16 

.10 

.42 

100 

2  00 

1  00 

0  20 

.15 

.58 

110 

2  12 

1  121 

0  24 

.19 

.77 

120 

2  24 

1  26J 

0  29 

.25 

1.00 

130 

2  36 

1  41i 

0  34 

.32 

1.28 

140 

2  48 

1  57i 

0  39 

.40 

1  60 

150 

3  00 

2  15 

0  45 

.49 

1.96 

160 

3  12 

2  33i 

0  51 

.59 

2.38 

170 

3  24 

2  53£ 

0  58 

.71 

2  86 

180 

3  36 

3  14* 

1  05 

.85 

3.39 

.1 

190 

3  48 

3  36i 

1  12 

1.00 

3  99 

.1 

200 

4  00 

4  00 

1  20 

1.16 

4  65 

.1 

.0 

100 


1°  in  50  ft. 


TABLE  VII.— Continued. 


0=2. 


Length 

D 

A 

e 

0 

y 

x  COR. 

*COR. 

210 

4°12' 

4°  241' 

1°28' 

1.35 

5.38 

.1 

0 

220 

4  24 

4  50^ 

1  37 

1.54 

6.19 

.2 

230 

4  36 

5  171 

1  46 

1.76 

7.07 

.2 

240 

4  48 

5  451 

1  55 

2.00 

8.04 

.2 

250 

5  00 

6  15 

2  05 

2.27 

9.09 

.3 

260 

5  12 

6  451 

2  15 

2.55 

10.22 

.4 

270 

5  24 

7  171 

2  26 

2.85 

11.45 

.4 

280 

5  36 

7  50i 

2  37 

3.18 

12.75 

.5 

290 

.5  48 

8  241 

2  48 

3  54 

14  18 

.6 

.1 

300 

6  00 

9  00 

3  00 

3  .91 

15.68 

.7 

.1 

310 

6  12 

9  361 

3  12 

4  32 

17.31 

.9 

.1 

320 

6  24 

10  141 

3  25 

4.75 

19.03 

1.0 

.2 

330 

6  36 

10  531 

3  38 

5  21 

20.87 

1.2 

.2 

340 

6  48 

11  33J 

3  51 

5.70 

22  81 

1.4 

.2 

350 

7  00 

12  15 

4  05 

6.22 

24.87 

1.6 

.3 

360 

7  12 

12  571 

4  19 

6.77 

27.05 

1.8 

.3 

370 

7  24 

13  411 

4  34 

7.34 

29.35 

2.1 

.3 

380 

7  36 

14  261 

4  49 

7.95 

31.79 

2.4 

.4 

390 

7  48 

15  12-1 

5  04 

8.60 

34.35 

2.7 

.4 

400 

8  00 

16  00 

5  20 

9.28 

37.04 

3.1 

.5 

410 

8  12 

16  481 

5  36 

10  00 

39.85 

3  5 

.6 

420 

8  24 

17  381 

5  53 

10.73 

42.79 

4  0 

.7 

430 

8  36 

18  291 

6  10 

11.53 

45.88 

44 

.7 

440 

8  48 

19  211 

6  27 

12.34 

49.14 

5.0 

.8 

450 

9  00 

20  15 

6  45 

13.20 

52.55 

5.6 

.9 

460 

9  12 

21  091 

7  03 

14.09 

56.05 

6.3 

1.0 

470 

9  24 

22  051 

7  21 

15.02 

59.73 

6.9 

1.2 

480 

9  36 

23  021 

7  40 

15.99 

63.55 

7.7 

1.3 

490 

9  48 

24  001 

8  00 

17.00 

67.55 

8.5 

1  4 

500 

10  00 

25  00 

8  19 

18.05 

71.72 

9.4 

1.6 

510 

10  12 

26  001 

8  39 

19.15 

76  00 

10.4 

1.7 

520 

10  24 

27  021 

9  00 

20.27 

80.04 

11.4 

1.9 

530 

10  36 

28  051 

9  21 

21.45 

85.08 

12  6 

2  1 

540 

10  48 

29  09i 

9  42 

22.68 

89.88 

13.8 

2.3 

550 

11  00 

30  15 

10  031 

23.96 

94.85 

15.1 

2.5 

560 

11  12 

31  211 

10  26 

25.27 

99.97 

16.5 

2  8 

570 

11  24 

32  29| 

10  48 

26.62 

105  19 

18  0 

3.0 

580 

11  36 

33  38| 

11  101 

28.01 

110.62 

19.6 

3.3 

590 

11  48 

34  481 

11  34 

29.48 

116.27 

21.3 

3.6 

600 

12  00 

36  00 

11  58 

30.97 

122.13 

23.2 

3  9 

101 


1°  in  40  ft. 


VIII.     TRANSITION  SPIRAL. 


length 

D 

A 

0 

0 

y 

x  COR. 

t  COR. 

10 

0°15' 

o°or 

0°00' 

.00 

.00 

.0 

.0 

20 

0  30 

0  03 

0  01 

30 

0  45 

0  07 

0  02 

.02 

40 

1  00 

0  12 

0  04 

.01 

.05 

50 

1  15 

0  19 

0  06 

.02 

.09 

60 

1  30 

0  27 

0  09 

.04 

.16 

70 

1  45 

0  37 

0  12 

.06 

.25 

80 

2  00 

0  48 

0  16 

.09 

.37 

90 

2  15 

1  01 

0  20 

.13 

.53 

100 

2  30 

1  15 

0  25 

.18 

.73 

110 

2  45 

1  31 

0  30 

.24 

.97 

120 

3  00 

1  48 

0  36 

.31 

1.25 

130 

3  15 

2  07 

0  42 

.40 

1.60 

140 

3  30 

2  27 

0  49 

.50 

2.00 

150 

3  45 

2  49 

0  56 

.61 

2.45 

160 

4  00 

3  12 

1  04 

.74 

2.97 

170 

4  15 

3  37 

1  12 

.89 

3.57 

180 

4  30 

4  03 

1  21 

1.06 

4.24 

.1 

190 

4  45 

4  31 

1  30 

1.25 

4.99 

.1 

200 

5  00 

5  00 

1  40 

1.45 

5.81 

.2 

210 

5  15 

5  31 

1  50 

1  68 

6.72 

.2 

220 

5  30 

6  03 

2  01 

1.93 

7.74 

.2 

230 

5  45 

6  37 

2  12 

2.20 

8.85 

.3 

240 

6  00 

7  12 

2  24 

2.51 

10.05 

.4 

250 

6  15 

7  49 

2  36 

2.84 

11.37 

.5 

.1 

260 

6  30 

8  27 

2  49 

3.19 

12.77 

.6 

.1 

270 

6  45 

9  07 

3  02 

3.57 

14.29 

.7 

.1 

280 

7  00 

9  48 

3  16 

3.98 

15.94 

.8 

.1 

290 

7  15 

10  31 

3  30 

4.42 

17.70 

1.0 

.2 

300 

7  30 

11  15 

3  45 

4.89 

19.59 

1.2 

.2 

310 

7  45 

12  01 

4  00 

5.40 

21.61 

1.4 

.2 

320 

8  00 

12  48 

4  16 

5.94 

23.76 

1.6 

.3 

330 

8  15 

13  37 

4  32 

6.51 

26.05 

1.9 

.3 

340 

8  30 

H  27 

4  49 

7.12 

28.46 

2.2 

.4 

350 

8  45 

15  19 

5  06 

7.77 

31.03 

2.5 

.4 

360 

9  00 

16  12 

5  24 

8.46 

33.74 

2.9 

.5 

370 

9  15 

17  07 

5  42 

9.18 

36.62 

3.3 

.5 

380 

9  30 

18  03 

6  01 

9.95 

39.64 

3.7 

.6 

390 

9  45 

19  01 

6  20 

10.75 

42.82 

4.3 

.7 

400 

10  00 

20  00 

6  40 

11.60 

46.16 

4.9 

.8 

410 

10  15 

21  01 

7  00 

12  47 

49.65 

5  5 

.9 

420 

10  30 

22  03 

7  21 

13.39 

53.28 

6.2 

1.0 

430 

10  45 

23  07 

7  42 

14.38 

57.10 

6  9 

1.2 

440 

11  00 

24  12 

8  04 

15.39 

61.12 

7.8 

1.3 

450 

11  15 

25  19 

8  26 

16.45 

65.32 

8.7 

1.5 

1°  in  30  ft. 


IX.     TRANSITION  SPIRAL. 


Length 

D 

A 

9 

o 

y 

x  COR. 

/fCOR. 

10 

0°20' 

o°or 

0°00' 

.00 

.00 

.0 

.0 

20 

0  40 

0  04 

0  01 

.00 

.01 

30 

1  00 

0  09 

0  03 

.01 

.03 

40 

1  20 

0  16 

0  05 

.02 

.06 

50 

1  40 

0  25 

0  08 

.03 

.12 

60 

2  00 

0  36 

0  12 

.05 

.21 

70 

2  20 

0  49 

0  16 

.08 

.33 

80 

2  40 

1  04 

0  21 

.12 

.50 

90 

3  00 

1  21 

0  27 

.18 

.71 

100 

3  20 

1  40 

0  33 

.24 

.97 

110 

3  40 

2  01 

0  40 

.32 

1.29 

120 

4  00 

2  24 

0  48 

.42 

1.68 

130 

4  20 

2  49 

0  56 

.53 

2.13 

140 

4  40 

3  16 

1  05 

.67 

2.66 

150 

5  00 

3  45 

1  15 

.82 

3.27 

.1 

160 

5  20 

4  16 

1  25 

.99 

3.97 

.1 

170 

5  40 

4  49 

1  36 

1.19 

4.76 

.1 

180 

6  00 

5  24 

1  48 

1.41 

5.65 

.2 

190 

6  20 

6  01 

2  00 

1.66 

6.65 

.2 

200 

6  40 

6  40 

2  13 

1.94 

7.75 

.3 

210 

7  00 

7  21 

2  27 

2.24 

8.97 

.3 

.1 

220 

7  20 

8  04 

2  41 

2.58 

10.31 

.4 

.1 

230 

7  40 

8  49 

2  56 

2.95 

11.77 

.5 

.1 

240 

8  00 

9  36 

3  12 

3.35 

13.38 

.7 

.1 

250 

8  20 

10  25 

3  28 

3.78 

15.11 

.8 

.1 

260 

8  40 

11  16 

3  45 

4.25 

17.00 

1.0 

.2 

270 

9  00 

12  09 

4  03 

4.76 

19.02 

1.2 

.2 

280 

9  20 

13  04 

4  21 

5.31 

21.20 

1.4 

.2 

290 

9  40 

14  01 

4  40 

5.90 

23.55 

1.7 

.3 

300 

10  00 

15  00 

5  00 

6.53 

26.05 

2.0 

.3 

310 

10  20 

16  01 

5  20 

7.20 

28.72 

2.4 

.4 

320 

10  40 

17  04 

5  41 

7.92 

31.57 

2.8 

.5 

330 

11  00 

18  09 

6  03 

8.69 

34.59 

3.3 

.5 

340 

11  20 

19  16 

6  25 

9.49 

37.80 

3.8 

.6 

350 

11  40 

20  25 

6  48 

10.35 

41.19 

4.4 

.7 

360 

12  00 

21  36 

7  11 

11.25 

44.78 

5.1 

.8 

370 

12  20 

22  49 

7  36 

12.21 

48.56 

5.8 

1.0 

380 

12  40 

24  04 

8  00 

13.22 

52.53 

6.6 

1.1 

390 

13  00 

25  21 

8  26 

14.28 

56.71 

7.6 

1.3 

400 

13  20 

26  40 

8  52 

15.39 

61.10 

8.6 

1.4 

410 

13  40 

28  01 

9  19 

16.56 

65.69 

9.7 

1.6 

420 

14  00 

29  24 

9  47 

17.79 

70.49 

10.9 

1.8 

430 

14  20 

30  49 

10  15 

19.07 

75.51 

12.3 

2.1 

440 

14  40 

32  16 

10  43 

20.41 

80.74 

13.7 

2.3 

450 

15  00 

33  45 

11  13 

21.81 

86.19 

15.4 

2.6 

TABLE  X.     TRANSITION  SPIRAL. 


1°  in  20  ft. 


a=5. 


Length 

D 

A 

0 

0 

y 

x  COR. 

/COR. 

~10~ 

0°30' 

o°or 

0°00' 

.00 

.00 

.0 

.0 

20 

1  00 

0  06 

0  02 

.01 

30 

1  30 

0  13 

0  04 

.01 

.04 

40 

2  00 

0  24 

0  08 

.02 

.09 

50 

2  30 

0  37 

0  12 

.05 

.18 

60 

3  00 

0  54 

0  18 

.08 

.31 

70 

3  30 

1  13 

0  24 

.12 

.50 

80 

4  00 

1  36 

0  32 

.19 

.74 

90 

4  30 

2  01 

0  40 

.26 

1.06 

100 

5  00 

2  30 

0  50 

.36 

1.45 

110 

5  30 

3  01 

1  00 

.48 

1.94 

120 

6  00 

3  36 

1  12 

.62 

2.51 

130 

6  30 

4  13 

1  24 

.79 

3.20 

140 

7  00 

4  54 

1  38 

.99 

3.99 

.1 

150 

7  30 

5  37 

1  52 

1.22 

4.90 

.1 

160 

8  00 

6  24 

2  08 

1.48 

5.96 

.2 

170 

8  30 

7  13 

2  24 

1.78 

7.15 

.3 

180 

9  00 

8  06 

2  42 

2.11 

8.49 

.4 

190 

9  30 

9  01 

3  00 

2.49 

9.98 

.5 

200 

10  00 

10  00 

3  20 

2.90 

11.62 

.6 

.1 

210 

10  30 

11  01 

3  40 

3.36 

13.45 

.8 

.1 

220 

11  00 

12  06 

4  02 

3.86 

15.44 

1.0 

.2 

230 

11  30 

13  13 

4  24 

4.41 

17.63 

1.2 

.2 

240 

12  00 

14  24 

4  48 

5.01 

20.01 

1.5 

.3 

250 

12  30 

15  37 

5  12 

5.66 

22.60 

1.8 

.3 

260 

13  00 

16  54 

5  38 

6.37 

25.38 

2.2 

.4 

270 

13  30 

18  13 

6  04 

7  12 

28  39 

2.7 

.5 

280 

14  00 

19  36 

6  32 

7.94 

31.62 

3.3 

.6 

290 

14  30 

21  02 

7  00 

8.82 

35.10 

3.9 

.7 

300 

15  00 

22  30 

7  29 

9.76 

38.83 

4.6 

.8 

310 

15  30 

24  02 

8  00 

10.76 

42.73 

5.4 

.9 

320 

16  00 

25  36 

8  31 

11.82 

46  92 

6.3 

1.1 

330 

16  30 

27  13 

9  04 

12.95 

51  36 

7  4 

1.2 

340 

17  00 

28  54 

9  37 

14.15 

56.05 

8.6 

1.4 

350 

17  30 

30  37 

10  11 

15.43 

61.09 

9.9 

1.7 

360 

18  00 

32  24 

10  46 

16.75 

66.31 

11  3 

1.9 

370 

18  30 

34  14 

11  19 

18.16 

71.63 

13.0 

2.2 

380 

19  00 

36  06 

12  00 

19.65 

77.35 

14.8 

2.5 

390 

19  30 

38  02 

12  38 

21.21 

83.41 

16  8 

2.8 

400 

20  00 

40  00 

13  17 

22.87 

89.83 

19.0 

3  2 

104 


TABLE  XI.     TRANSITION  SPIRAL. 


1°  in  10  ft. 


a=10. 


Length 

D 

A 

6 

0 

y 

JfCOR. 

*COR. 

10 

1°00' 

0°03' 

o°or 

.00 

.00 

.0 

.0 

20 

2 

0  12 

0  04 

.01 

.02 

30 

3 

0  27 

0  09 

.02 

.08 

40 

4 

0  48 

0  16 

.05 

.19 

50 

5 

1  15 

0  25 

.09 

.36 

60 

6  00 

1  48 

0  36 

.16 

.63 

70 

7 

2  27 

0  49 

.25 

1.00 

80 

8 

3  12 

1  04 

.37 

1.49 

90 

9 

4  03 

1  21 

.53 

2.12 

100 

10 

5  00 

1  40 

.73 

2.91 

.1 

110 

11  00 

6  03 

2  01 

.97 

3  87 

.1 

120 

12 

7  12 

2  24 

1.26 

5.02 

.2 

130 

13 

8  27 

2  49 

1.60 

6.38 

.3 

140 

14 

9  48 

3  16 

1.99 

7.97 

.4 

.1 

150 

15 

11  15 

3  45 

2.45 

9.79 

.6 

.1 

160 

16  00 

12  48 

4  16 

2.97 

11.87 

.8 

.1 

170 

17 

14  27 

4  49 

3.56 

14.23 

1.1 

.2 

180 

18 

16  12 

5  24 

4.23 

16.87 

1.4 

.2 

190 

19 

18  03 

6  01 

4  97 

19.81 

1  9 

.3 

200 

20 

20  00 

6  39 

5.79 

23.07 

2.4 

.4 

210 

21  00 

22  03 

7  20 

6.70 

26.65 

3.1 

.5 

220 

22 

24  12 

8  03 

7.69 

30.58 

3.9 

.6 

230 

23 

26  27 

8  48 

8.78 

34.86 

4.8 

.8 

240 

24 

28  48 

9  35 

9.96 

39.49 

6  0 

1.0 

250 

25 

31  15 

10  23 

11.24 

44.49 

!     7.3 

1.2 

TABLK  XII.  FACTORS  FOR  ORDINATKS, 

To  find  y,  multiply  o  by  the  factor  for  the  ratio  found  by 
dividing  the  distance  of  the  point  from  the  P.  S.  or  P.  C.  C. 
by  the  half-length  of  spiral. 


Ratio  to 
#  length 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1  0 

Factor... 

.0005 

.004 

.014 

032 

.063 

108 

172 

.256 

.365 

.500 

105 


TABLE  XIII. 

SPIRAL  DEFLECTION  ANGLES   FOR  FIRST 
CHORD  LENGTH. 

Values  of  9l  in  minutes  for  use  with  Table  XIV. 

The  first  line  of  captions  gives  the  distance  in  feet  in  which  1°  of  degree- 
of-curve  is  attained.  The  second  line  of  captions  gives  the  value  of  a. 
Angles  are  given  in  minutes. 


Chord 

200 

150 

125 

100 

80 

75 

Chord 

Length 

/2 

2A 

4/5 

1 

IX 

IK 

Length 

10 

.050 

.067 

.080 

.100 

.125 

.133 

10 

11 

.060 

.081 

.097 

.121 

.151 

.161 

11 

12 

.072 

.096 

.115 

.144 

.180 

.192 

12 

13 

.084 

.113 

.135 

.169 

.211 

.225 

13 

14 

.088 

.131 

.157 

.196 

.245 

.261 

14 

15 

.112 

.150 

.180 

.225 

.281 

.300 

15 

16 

.128 

.171 

.205 

.256 

.320 

.341 

16 

17 

.144 

.193 

.231 

.289 

.361 

.385 

17 

18 

.162 

.216 

.259 

.324 

.405 

.432 

18 

19 

.180 

.241 

.289 

.361 

.451 

.481 

19 

20 

.200 

.267 

.320 

.400 

.500 

.533 

20 

25 

.312 

.417 

.500 

.625 

.781 

.833 

25 

30 

.450 

.600 

.720 

.900 

1.125 

1.200 

30 

35 

.612 

.817 

.980 

1.225 

1.531 

1.633 

35 

40 

.800 

1.067 

1.280 

1.600 

2.000 

2.133 

40 

45 

1.012 

1.350 

1.620 

2.025 

2.531 

2.700 

45 

50 

1.250 

1.667 

2.000 

2.500 

3.125 

3.333 

50 

Chcrd 

60 

50 

40 

30 

25 

20 

Chord 

Length 

1% 

2 

2/2 

3K 

4 

5 

Length 

10 

.167 

.200 

.250 

.333 

.400 

.500 

10 

11 

.202 

.242 

.302 

.403 

.484 

.605 

11 

12 

.240 

.288 

.360 

.480 

.576 

.720 

12 

13 

.282 

.338 

.422 

.563 

.676 

.845 

13 

14 

.327 

.392 

.490 

.653 

.784 

.980 

14 

15 

.375 

.450 

.562 

.750 

.900 

1.125 

15 

16 

.427 

.512 

.640 

.853 

1.024 

1.280 

16 

17 

.482 

.578 

.722 

.963 

1.156 

1.445 

17 

18 

.540 

.648 

.810 

1.080 

1.296 

1.620 

18 

19 

.602 

.722 

.902 

1.203 

1.444 

1.805 

19 

20 

.667 

.800 

1.000 

1.333 

1.600 

2.000 

20 

25 

1.042 

1.250 

1.562 

2.083 

2.500 

3.125 

25 

30 

1.500 

1.800 

2.250 

3.000 

3.600 

4.500 

30 

35 

2.042 

2.450 

3.062 

4.083 

4.900 

6.125 

35 

40 

2.667 

3.200 

4.000 

5.333 

6.400 

8.000 

40 

45 

3.375 

4.050 

5.062 

6.750 

8.100 

10.125 

45 

50 

4.667 

5.000 

6.250 

8.333 

10.000 

12.500 

50 

•Hi 


a 


a  I 


4.     § 


fl  "a 

H 

,2    a 


ai  "8 


w& 

<=>-«««"« 

rH 

rHrHrHrHi—  I 

10 

rH 

338358 

CM  ^  VO  CO  O 
CO  CM  CM  CM  CM 

rt-  vo  VO  rf- 
VO  CM  CO  Tt<-N 
rH  rH                  ^ 

rH 

CM  !>•  O  rH  O  l~>» 

CM  VO  VO  vo  CM 
Xs*  ^  rH  CO  vo 
CM  CM  CM  rH  rH 

t^    O    CO              r-- 

rH  CO  ^  Q  ^t" 

CO 
rH 

00  Tf  00  O  O  00 
CO  CO  CO  CM  CM  CM 

^  OO  O  O  CO 
CM  ON  t^  ^1"  O 
CM  -*  rH  r-i   rH 

rf  CO        O  CM 

CM 

rH 

CO  vo  O  CO  Tj"  CO 

co  r^  vo  Tt  CM  o 

CM  CM  Cxi  CM  CM  CM 

O  10  CO  ON  CO 
CO  vo  CM  O\  VO 
rH  rH  rH 

vo       r>-  vo  r^- 
^o^  ^^ 

rH 
rH 

CM  O  VO  O  CM  CM 
•^-  CO  rH  O  CO  VO 
CM  CM  CM  CM  rH  rH 

O  VO  O  CM  CM 
Tt  rH  O\  VO  CO 
rH  rH 

Tf  O  00  CO 
rH  rH 

w 

3       o 

SON  VO  rH  Tfr  10 
CO  t^>  VO  '^t"  CM 

rf  rH  VO  O> 
O  00  10  CMQ 

rH  rj-  O>  VO  vo 
CO  VO  ON  co  t^ 

S            rH 

P 

H        o> 

CM  CM  O  VO  O  CM 

CM  O  VO         00 

00  O  Tf  O  00 
VO  ON  CM  v<?  O^ 

fc 

M 

Q            CO 

3 

0 

CO  O^  CO  IO  OcO 
CM  rH  O  <3\  00  VO 
rH  rH  rH 

Th  co         vo  CM 

f-H  CM  vo  O  t*^ 
00  rH  Tf  00  rH 
rH  rH  rH  CM 

W 
< 

00  O  O  00  -^  CO 
CTs  ON  CO  VD  IO  CO 

O         CM  VO  CM 

8OCM  VO  CM 
CO  VO  C^  co 
rH  rH  rH  rH  CM 

H 

fc                      CO 
g 

8 

CM  IO  VO  IO  CM  l^ 

O'N  O  CO  CO 
Q'-l  "T  VO  CO 

VO  H-  IO  CO  CO 
rH  rH  rH  CM  CM 

P 

tf 

H            10 
05 

O  rj-  VO  VO  rl- 
»0  Tf  CO  CM  rHo 

VO  rj-  ^t  VO  O 

rH  CO  VO  t^-  O 
rH 

VO  Tfrj-  VO  O 

Cxi  VO  00  rH  VO 
rH  rH  rH  CM  CM 

CM  t^  O  rH         CO 
CO  CM  CM  rH  QrH 

89S8§ 

CO  O  O>  O  co 
CO  VO  00  CM  vo 
rH  rH  rH  CM  Cxi 

CO 

CO  ^-00         O  CM 
rH  rH          Q  rH  CM 

VO  CM  O  O  CM 
rH 

VO  Cxi  O  O  CM 
CO  VO  O^  CM  VO 
rH  rH  t-H  CM  CM 

CM 

o     rH(NI 

O  vo  CM  rH  CM 
rH 

VO  O  t^  VO  t^ 

CO  VO  CO  •—  t  rf 
rH  rH  rH  Cxi  Cxi 

rH 

CM         TfO  CO  00 
Q          rH  rH  CM 

o  ^t-o  oo  oo 

rl-  vo  t^  CO  O 

O  «fr  O  00  00 
CO  uo  CO  O  CO 
rH  rH  rH  CM  Cxi 

O 

rH  Tt  O\  VO  <O 

0 

VO  O>  ^f  rH  O 

co  ^t  vo  oo  o 

rH 

rH  Tt  O\  VO  VO 
CM  rj-  VO  ON  CM 
rH  rH  rH  rH  CM 

jnio.1  pjoqa 

01-10300^10 

rH 

rH  CM  COTf*  10 

TABLE  XV. 


£=2000. 


STREET  RAILWAY  SPIRAL. 

£=2000. 


Length 

Radius 

J 

0 

o 

y 

.arCoR. 

/COR. 

5 

400 

0°21f 

0°07' 

.01 

.00 

.00 

10 

200 

1  26 

0  29 

.08 

.00 

15 

133.33 

3  13 

1  04 

.28 

.00 

16 

125 

3  40 

1  13 

.34 

.01 

20 

100 

5  44 

1  55 

.17 

.67 

.02 

21.06 

95 

6  21 

2  07 

.19 

.78 

.03 

22.22 

90 

7  04 

2  21 

.23 

.91 

.03 

.00 

23.53 

85 

7  56 

2  39 

.27 

1.09 

.04 

.01 

25 

80 

8  57 

2  59 

.32 

1.30 

.06 

.01 

26.67 

75 

10  11 

3  24 

.39 

1.58 

.08 

.01 

1 

28.57 

70 

11  41 

3  54 

.48 

1.94 

.12 

.02 

30 

66.67 

12  54 

4  18 

.56 

2.24 

.15 

.02 

30.77 

65 

13  34 

4  31 

.61 

2.42 

.17 

.03 

33  33 

60 

15  55 

5  18 

.77 

3.07 

.26 

.04 

36.36 

55 

18  56 

6  19 

1  00 

3.98 

.40 

.07 

40 

50 

22  55 

7  38 

1  32 

5  27 

.64 

.11 

44.44 

45 

28  17 

9  25 

1.81 

7.19 

1.08 

.18 

50 

40 

35  49 

11  54 

2.56 

10.13 

1.95 

.32 

55 

36.36 

43  20 

14  23 

3.39 

13.29 

3  15 

.52 

56.05 

35.68 

45  00 

14  55 

3.59 

14  02 

3  46 

.58 

TABLE  XVI.     STREET  RAILWAY  SPIRAL. 
£=1500.  £=1500. 


Length 

Radius 

J 

9 

o 

.1 

y 

.r  COR. 

/COR. 

5 

300 

0°29' 

0°10' 

.01 

.00 

.00 

10 

150 

1  55 

0  38 

.11 

.00 

12 

125 

2  45 

0  55 

.19 

.CO 

15 

100 

4  18 

1  26 

.38 

.01 

15.79 

95 

4  46 

1  35 

.43 

.01 

16.67 

90 

5  19 

1  46 

.51 

.01 

17.65 

85 

5  58 

1  59 

.61 

.02 

18.75 

80 

6  43 

2  14 

.18 

.73 

.03 

20 

75 

7  38 

2  33 

.22 

.89 

.03 

.00 

21  A3 

70 

8  46 

2  55 

.27 

1.09 

.05 

.01 

23.08 

65 

10  10 

3  23 

.34 

1.36 

.07 

.01 

25 

60 

11  56 

3  59 

.43 

1.74 

.11 

.02 

27.27 

55 

14  13 

4  44 

.56 

2.24 

.17 

.03 

30 

50 

17  11 

5  44 

.75 

2.98 

.27 

.04 

33.33 

45 

21  13 

7  04 

1.02 

4.07 

.45 

.07 

35 

42.86 

23  24 

7  47 

1.18 

4.71 

.57 

.09 

37.50 

40 

26  51 

8  56 

1.45 

5.77 

.82 

.14 

40 

37.50 

30  34 

10  10 

1.76 

6.96 

1.14 

.19 

42.86 

35 

35  05 

11  40 

2.16 

8.51 

1.61 

.27 

45 

33.33 

38  41 

12  51 

2.49 

9.80 

2.05 

.34 

TABLE  XVII.     STREET  RAILWAY  SPIRAL. 
£-:1250  £  =  1250 


Length 

Radius 

A 

9 

o 

V 

x  COR. 

/COR. 

5 

250 

0°34' 

0°11' 

.02 

.00 

.00 

10 

125 

2  17 

0  46 

.13 

15 

83.33 

5  11 

1  44 

.45 

.01 

15.62 

80 

5  36 

1  52 

.51 

.01 

16.67 

75 

6  22 

2  07 

.15 

.62 

.02 

17.86 

70 

7  19 

2  26 

.19 

.76 

.03 

.00 

19.23 

65 

8  29 

2  50 

.24 

.95 

.04 

.01 

20 

62,50 

9  10 

3  03 

.26 

1.04 

.05 

.01 

20.83 

60 

9  56 

3  19 

.30 

1.20 

.06 

.01 

22.73 

55 

11  50 

3  57 

.39 

1.56 

.10 

.02 

25 

50 

14  19 

4  46 

.52 

2  07 

.15 

.02 

27.  78 

45 

17  41 

5  54 

.71 

2.84 

26 

.04 

30 

41  66 

20  38 

6  52 

.89 

3.56 

38 

.06 

31.25 

40 

22  24 

7  28 

1.01 

5.04 

.47 

.08 

35 

35.71 

28  05 

9  20 

1.40 

5.62 

.83 

.14 

TABLE  XVIII. 

£--  -1000 


STREET  RAILWAY  SPIRAL. 

£  =  1000 


Length 

Radius 

A 

a 

o 

y 

x  COR. 

/COR. 

5 

200 

0°43' 

0°14' 

.02 

.00 

.00 

10 

100 

2  52 

0  57 

.17 

.01 

15 

66.67 

6  27 

2  09 

.56 

.02 

15.39 

65 

6  47 

2  16 

.15 

.61 

.02 

16.67 

60 

7  57 

2  39 

.19 

.77 

.03 

.00 

18.18 

55 

9  28 

3  09 

.25 

1.00 

.05 

.01 

20 

50 

11  27 

3  49 

.33 

1  33 

.08 

.01 

22.22 

45 

14  09 

4  44 

.46 

1.83 

.13 

.02 

25 

40 

17  54 

5  57 

.64 

2.58 

.24 

.04 

28.57 

35 

23  23 

7  47 

.97 

3.84 

.47 

.08 

30 

33.33 

25  47 

8  34 

1.12 

4.45 

.60 

.10 

33.33 

30 

31  50 

10  35 

1.53 

6.05 

1.03 

.17 

TABLE  XIX.     STREET  RAILWAY  SPIRAL, 


=  750 


£=750 


Length 

Radius 

A 

8 

o 

y 

x  COR. 

/COR. 

5 

150 

0°57' 

0°19' 

.03 

.00 

.00 

10 

75 

3  49 

1  16 

.22 

.01 

15 

50 

8  54 

2  52 

.19 

.75 

.03 

.00 

16.67 

45 

10  37 

3  32 

.26 

1.03 

.06 

.01 

18.75 

40 

13  24 

4  28 

.36 

1.46 

.10 

.02 

20 

37.5 

15  17 

5  05 

.44 

1.75 

.14 

.02 

21.43 

35 

17  32 

5  50 

.57 

2.27 

.20 

.03 

25 

30 

23  52 

7  57 

.86 

3.43 

.43 

.07 

TABLE  XX.     OFFSETS  FOR  SPIRALS. 


T) 

3° 

4° 

5° 

«. 

T° 

a 

o 

L 

0 

L 

o 

L 

o 

L 

o 

L 

0  5 

7  83 

6.000 

0  6 

5  44 

5.000 

12.86 

6.667 

0.7 

4  00 

4  286 

9-47 

5.714 

18.47 

7.143 

0-8 

3-06 

3.750 

7.24 

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